V 


APPLETONS 


CYCLOPAEDIA  OF  DRAWING, 


DESIGNED   AS 


E  KK  A  T  A. 

In  "  Kule  and  Examples,"  pp.  223  and  224,  rule  should  be : 

"  Multiply  one-half  the  weight  of  the  rafter  and  the  weight  distributed 
on  it  by  one-half  the  span,  and  divide  the  product  by  the  pitch." 

In  example,  weight,  instead  of  "  8,500  Ibs.,"  should  be  "  4,250  Ibs.," 
and  result  "  0.708  square  inches." 

P.  224,  instead  of  "40x31x10=12,400  Ibs.,"  should  be  "40x35x 
10=14,000,"  and  corresponding  changes  in  rest  of  calculations. 


NEW    AND     ENLARGED     ED 


YOKK: 
D.     APPLETON     AND     COMPANY, 

90,    92    &    94    GRAND    STREET. 

1869. 


APPLETONS' 


CYCLOPEDIA  OF  DRAWING, 


DESIGNED   AS 


A  TEXT-BOOK 


MECHANIC,  ARCHITECT,  ENGINEER,  AND  SURVEYOR, 


COMPRISING 


GEOMETRICAL  PROJECTION,  MECHANICAL,  ARCHITECTURAL,  AND  TOPOGRAPHICAL 
DRAWING,  PERSPECTIVE  AND  ISOMETRY. 


EDITED    BY 

W.    E.    WORTHED. 

NEW    AND     ENLARGED     EDITION, 


NEW  YOKK: 
D.     APPLETON     AND     COMPANY, 

90,    92    &    94    GRAND    STREET. 
1869. 


ENTERED,  according  to  Act  of  Congres?,  in  the  year  1857,  by 

D.  APPLETOX  &  CO., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Southern 
District  of  New  York. 


ENTERED,  according  to  Act  of  Congress,  in  the  year  186S,  by 

D.  APPLETON  &  CO., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Southern 
District  of  New  York. 


PEEFAOE. 

AT  the  suggestion  of  the  publishers,  this  work  was  undertaken  to 
form  one  of  their  series  of  Dictionaries  and  Cyclopaedias.  In  this  view, 
it  has  been  the  intention  to  make  it  a  complete  course  of  instruction 
and  book  of  reference  to  the  mechanic,  architect  and  engineer.  It 
has  not,  therefore,  been  confined  to  the  explanation  and  illustration 
of  the  methods  of  projection,  and  the  delineation  of  objects  which 
might  serve  as  copies  to  the  draughtsman,  matters  of  essential  impor- 
tance for  the  correct  and  intelligible  representation  of  every  form ;  but 
it  contains  the  means  of  determining  the  amount  and  direction  of 
strains  to  which  different  parts  of  a  machine  or  structure  may  be  sub- 
jected, and  the  rules  for  disposing  and  proportioning  of  the  material 
employed,  to  the  safe  and  permanent  resistance  of  those  strains,  with 
practical  applications  of  the  same.  Thus  while  it  supplies  numerous 
illustrations  in  every  department  for  the  mere  copyist,  it  also  affords 
suggestions  and  aids  to  the  mechanic  in  the  execution  of  new  designs. 
And  although  the  arranging  and  properly  proportioning  alone  of  mate- 
rial in  a  suitable  direction  and  adequately  to  the  resistance  of  the  strains 
to  which  it  might  be  exposed,  would  produce  a  structure  sufficient  in 
point  of  strength  for  the  purposes  for  which  it  is  intended,  yet  as  in 
many  cases  the  disposition  of  the  material  may  be  applied  not  only 
practically,  but  also  artistically,  and  adapted  to  the  reception  of  orna- 
ment, under  the  head  of  Architectural  Drawing,  the  general  charac- 
teristics of  various  styles  have  been  treated  of,  and  illustrated,  with 
brief  remarks  on  proportion  and  the  application  of  color. 


Within  the  last  few  years,  both  here  and  abroad,  a  number  of  works 
have  been  published  on  "  Practical  Drawing,"  but  no  one  work  has  illus- 
trated all  departments  of  the  sitbject.  In  the  mechanical,  the  works  of 
M.  Le  Brun  and  M.  M.  Arrnengaud  are  the  standard  which  have  been 
made  the  basis  of  two  English  works,  "  The  Practical  Draughtsman's 
Book  of  Industrial  Design"  and  the  "Engineer's  and  Machinist's  Draw- 
ing Book."  From  the  latter  of  these  works  we  have  drawn  most  of 
our  chapters  on  Geometrical  and  Mechanical  Drawing,  and  Shades  and 
Shadows.  In  neither  the  French  nor  the  English  works  has  the  science 
of  architectural  construction  and  drawing  been  adequately  illustrated, 
nor  has  Topographical  Drawing  been  treated  of.  In  these  two 
departments  a  varied  selection  has  been  made  from  the  best  authorities. 
In  the  Architectural,  Ferguson  and  Garbett  have  been  the  most  con- 
sulted; in  the  Topographical,  Williams,  Gillespie,  Smith,  and  Frome. 
The  work  will  be  found  quite  fully  illustrated,  and  the  drawings  and 
engravings  have  been  carefully  executed,  mostly  under  the  supervision  of 
Mr.  H.  Grassau. 

Like  most  cyclopaedias,  this  work  claims  for  its  articles  but  little  of 
novelty  or  originality ;  the  intention  of  the  compiler  was,  to  collect  within 
moderate  compass  as  much  valuable  matter  as  possible,  in  practical  Draw- 
ing and  Design  ;  and  to  this  purpose  he  brings  the  experience  of  series  of 
years  in  each  of  the  departments  treated.  Practically,  he  has  had  means 
of  knowing  the  necessities  of  the  trade  and  of  the  profession,  and  trusts 
that  the  selection  now  made  will  be  found  useful  for  the  purposes  for 
which  it  was  intended. — W. 


TABLE    OF    CONTENTS. 


GEOMETRICAL  DEFINITIONS  AND  TECHNICALI- 
TIES, 1-6. 

DRAWINO  INSTRUMENTS. — Description  and 
use;  pencil;  ruler;  triangle;  T  square; 
parallel  ruler ;  sweeps  and  variable  curves, 
compasses,  or  dividers;  drawing  pens; 
dotting  point ;  drawing  pins ;  scales ;  pro- 
tractor ;  vernier  scales ;  scale  of  rhumbs, 
longitude,  chords,  sines,  tangents ;  the  sec- 
tor; Marquois's  scales;  triangular  com- 
passes ;  wholes  and  halves ;  beam  com- 
passes; portable  compasses;  screw  di- 
viders ;  circular  protractor ;  pentograph  ; 
camera  lucida ;  drawing  table  and  draw- 
ing board,  7-35. 

Drawing  paper ;  tracing  paper ;  mouth 
glue;  mounting  paper  and  drawings; 
varnishing;  management  of  instruments, 
35^4. 

GEOMETRICAL  PROBLEMS. — Drawing  of  lines, 
division  of  lines,  perpendiculars  to,  par- 
allels; construction  of  angles;  division  of 
angles;  description  of  arcs  and  circles; 
connection  of  straight  lines  by  arcs,  and 
arcs  with  arcs ;  compound  curves,  42-56. 

On  circles  and  rectilinear  figures;  triangles, 
squares,  rectangles,  parallelograms;  in- 
scribed and  described  circles ;  pentagons ; 
hexagons,  octagons,  polygons;  table  of 
polygonal  angles,  56-63. 

On  the  use  of  the  T  square,  and  triangle  in 
construction  of  preceding  problems;  di- 
vision of  lines,  63-05. 

Simple  application  of  regular  figures,  65. 

Problems  on  proportional  lines  and  equiv- 
alent figures,  66. 

On  the  ellipse,  parabola,  hyperbola,  cycloid, 
epicycloid,  involute,  and  spiral,  68-79. 


|  GEOMETRICAL  PROJECTION. — Of  the  point, 
line,  solids;  plans,  elevations,  sections, 
80-82. 

Shade  lines  in  outline  drawings,  83. 

Projections  of  simple  bodies;  hexagonal 
pyramid ;  prism,  83-88. 

Construction  of  conic  sections,  90. 

Penetrations  or  intersections  of  solids;  of 
cylinders,  cones,  and  prisms;  cylinders, 
prisms,  spheres,  and  cones,  92-99. 

Of  the  helix,  99. 

Development  of  surfaces,  cylinder,  cone, 
sphere,  102. 

MECHANICS. — The  mechanical  powers;  the 
lever,  wheel,  and  axle,  pulley,  inclined 
plain,  wedge,  screw,  105-110. 

Forces,  parallel,  inclined,  parallelogram  of, 
composition  and  resolution  of,  centre  of 
gravity,  110-114. 

Friction  and  limiting  angle  of  resistance,  ex- 
periments by  M.  Moriri,  115-117. 

Equilibrium  of  the  polygon  of  rods  or  cords, 
application  to  framing,  117-119. 

Mechanical  properties  of  materials;  tables 
of  strength  of  woods,  of  metals;  resist- 
ance to  compression,  bricks,  granite,  cast- 
iron  pillars  ;  tensile  strength  ;  transverse 
strength,  beams,  girders  ;  detrusion ;  ten- 
sion, 120-129. 

Mechanical  work  or  effect,  of  animals,  of 
water,  of  steam ;  the  indicator ;  effect  of 
expansion;  table  of  pressure,  temperature 
and  volumes  of  steam;  table  of  weights 
and  evaporative  power  of  different  fuels  ; 
determination  of  water,  fuel,  and  size  of 
boiler  to  produce  a  given  power,  130-136. 

DBA  WING  OF  MACHINERY.  —  Shafting,  sec- 
tions of  wooden,  cast  and  wrought  iron ; 


TABLE    OF    CONTEXTS. 


table  of  diameters  of  journals;  tranverse 
strain;  water-wheel  shafts;  section  of 
water-wheel ;  crank  shaft  of  steam  engine ; 
table  of  diameters  of  journals  for  torsional 
strain ;  line  shafts,  137-142. 

Bearings  or  supports  for  journals;  steps; 
suspension  bearing  of  turbine,  step  or 
guide  for  same;  pillow-block  standard; 
side,  sprawl,  yoke  hangers;  couplings, 
face,  sleeve,  screw,  clamp;  horned,  slide, 
or  clutch,  bayonet,  and  friction  cone ;  pul- 
leys, plate,  plain  and  curved  arms,  faced 
coupling;  drums,  wooden;  cone;  belts, 
table  of  strain  on,  strength  of;  fast  and 
loose  pulleys ;  oblique  shafts,  142-156. 

Gearing;  spur,  bevel-wheels ;  internal  gear, 
rack  and  pinions;  trundle  gear;  trans- 
mission of  motion  ;  size  of  gear  ;  pitch ; 
table  of  pitch,  diameter  and  number  of 
teeth ;  form  and  proportions  of  teeth  ;  by 
scale  ;  table  of  stress  at  pitch  circle,  thick- 
ness of  teeth  and  pitch ;  table  of  pitch  ; 
thickness,  length,  and  breadth  of  teeth  and 
velority ;  fundamental  principle  of  toothed 
wheels;  epicycloidal  teeth;  the  trundle, 
templates,  involute  teeth,  156-175. 

Projections  of  a  spur-wheel;  of  a  bevel- 
wheel  ;  of  a  skew  bevel ;  of  a  pinion  driv- 
ing a  rack ;  of  a  rack  driving  a  pinion ;  of 
a  wheel  and  tangent,  or  endless  screw; 
internal  spur-wheel  driving  a  pinion;  an 
internal  driven  by  a  pinion ;  eccentrics, 
175-189. 

Drawing  of  screws;  triangular  threaded 
screw  and  nut ;  square  threaded  screw  and 
nut ;  table  of  diameters  of  bolts  and  nuts 
and  threads  per  inch,  189-191. 

Hooks,  form  and  proportions  of,  192. 

Frames  of  cam-punch,  and  shear ;  of  plan- 
ing machine;  jack-screw;  hydraulic-press, 
action  of  same ;  frames  of  American 
marine  engines ;  working  beams,  Ameri-  j 
can  and  English  with  details ;  cranks,  pro- ! 
portion  of  eyes ;  connecting  rods  with 
details,  192-198. 

Location  of  machines ;  example  of  weaving  j 
rooms,  189-200. 

Machines;  marine  engines,  and  locomotives, 
in  skeleton  drawings ;  cataract  of  a  Cor-  j 
nish    engine ;    details    of   48    stop    gate  j 
Brooklyn    "Water-works ;     sections   of    a 
locomotive  boiler  ;  elevation  and  sections  ; 


of  engine  of  Golden  Gate ;  elevation  and 
sections  of  a  Lowell  turbine,  with  rules 
for  describing  curves  and  proportioning 
turbines,  201-208. 

AECUITECTUKAL  DRAWINGS. — Foundations, 
walls,  bond  of  and  thickness  of;  extract 
from  London  and  Liverpool  building  acts ; 
mortar;  arches,  209-216. 

Framing,  beams,  flooring,  bridging,  girders; 
size  of  joists ;  stirrup  irons ;  floors ; 
trussed  beams;  fire-proof  floors;  parti- 
tions ;  roofs,  pitch,  form  for  various  span, 
size  and  proportions  of  parts;  table  of 
same;  joints;  varieties  of  roofs,  hipped, 
gambrel  or  Mansard,  circular :  eaves ;  iron 
roofs,  details  of  one  ;  Crystal  Palace 
girders,  cast  and  wrought  iron;  princi- 
ples of  bracing;  use  of  counters;  truss 
by  tension  rod;  system  of  suspension 
truss  ;  a  completely  braced  frame  ;  bridge 
trusses,  216-234. 

Size  and  proportion  of  rooms ;  dining  rooms ; 
parlors,  drawing,  and  bed  rooms,  pantries; 
passages,  height  of  stories;  details  of  parts; 
stairs ;  doors ;  windows ;  bases  and  surbase- ; 
cornices;  fireplaces;  privies;  water  closets 
and  outhouses;  cess  pools;  wood  and  coal 
sheds,  234-245. 

Drawing,  applications  of,  to  the  laying  out  of 
house;  plans,  elevations,  and  section  of  a 
house ;  plans  of  familiar  forms  of  houses, 
245-250. 

Mouldings,  Greek  and  Roman;  orders  of- 
architecture,  with  examples ;  Tuscan, 
Doric,  Ionic,  Corinthian,  and  Composite ; 
Arcades ;  Romanesque  and  Gothic  mould- 
ings, jamb,  ai'ch;  capitals,  string  courses; 
cornices ;  arches,  semicircular,  segmen- 
tal,  stilted,  horse-shoe,  pointed  '  ogee, 
Tudor,  and  foiled  ;  domes  and  vaults  ; 
Byzantine,  Roman,  Gothic ;  buttresses  ; 
towers;  pinnacles;  spires.  Windows,  Ro- 
manesque, Byzantine,  Norman,  Gothic; 
doorways  ;  the  Renaissance,  Florentine, 
Venetian ;  ornament,  Greek,  Roman,  By- 
zantine, Romanesque,  Saracenic,  Gothic, 
Renaissance ;  balustrades,  250-279. 

Elevations  of  Houses,  city  and  country; 
details  of  windows  ;  verge-boards ;  chim- 
ney-tops ;  balcony ;  stables ;  city  tene- 
ment house  ;  stores  and  warehouses  ; 
School-houses  and  furniture  ;  Lecture 


TABLE   OF   CONTENTS. 


rooms,  Churches,  Theatres,  Legislative 
Halls ;  transmission  of  sound ;  space  re- 
quired for  seats;  size  of  pews;  require- 
ments of  churches;  examples  from  city 
practice ;  requirements  of  theatres ;  di-  j 
mensions  of  several ;  New  York  Crystal  j 
Palace,  280-299. 

Material  for  building ;  appropriate  color ;  i 
ventilation  and  warming ;  air  required  for  j 
respiration,  lighting  and  heating;  circu- 
lation; table  of  grains  of  moisture  in  1 
cubic  foot  of  air ;  methods  of  heating  fire- 
places, stovesx  hot-air  furnaces,  steam  and 
hot  water,  circulation;  ventilators,  299- 
306. 

Drainage;  sewer  pipes;  privy  vaults ;  light- 
ing; water  supply;  wells  and  water  pipes, 
306-308. 

Principles  of  Design ;  Extracts  from  Fergu- 
son's Hand-Book  of  Architecture,  the 
Encyclopaedia  Britannica;  the  two  great 
principles  of  art,  308-312. 

Shading  and  Shadows.— Diffusion  of  light ; 
direct  and  cast  shadows;  problems  for  de- 
termining the  outline  of  shadows,  by  a 
straight  line  upon  plain  and  curved  sur- 
faces, by  a  circle,  by  a  hexagonal  pyramid ; 
the  limit  of  shade  and  shadow  of  a  cylin- 
der; reversed  cone ;  a  prism;  the  shadows 
cast  in  the  interior  of  a  cylinder,  of  a  hemi- 
sphere, of  a  niche  ;  the  line  of  shade  in  a 
sphere,  and  its  shadow  on  a  plane;  line 
of  shade  on  the  surface  of  a  ring ;  the  out- 
lines of  shadows  cast  on  surfaces  of  screws 
and  nuts,  triangular  and  .square,  threaded, 
313-328. 

Manipulation  of  shades  and  shadows;  me- 
thods of  tinting  surfaces'  in  the  light,  in 
the  shade ;  shading  by  flat  tints ;  by 
softened  tints ;  elaboration  of  shading  and 
shadows;  depth  of  shadows ;  examples  of 
finished  shading,  328-338. 

Finished  coloring ;  color  of  materials ;  prep- 
aration of  tints  ;  body  color  ;  manipula- 
tion, washing,  or  sponging;  color  for 
wrought  iron,  brass,  copper  ;  intensity  of 
shades  and  shadoAvs ;  margin  of  light ;  ad- 
vantage of  washing ;  conventional  tints  for 
materials,  339-348. 

TOPOGRAPHICAL  DRAWING.  —  Conventional 
signs  and  representation  of  features  of  a 
country;  distinctive  marks  for  edifices, 


for  metals ;  methods  of  representing  hills, 
vertical  and  horizontal  slopes  by  a  scale 
of  shade  ;  contours,  339-354. 

Plotting ;  rough  sketch ;  choice  of  scale,  and 
scales  prescribed  by  different  commis- 
sions ;  lines  of  survey  ;  variation  of  needle ; 
survey  by  compass  and  plot ;  balancing  of 
error;  plotting  by  latitudes  and  depar- 
tures ;  plotting  of  offsets  by  scale  ;  plot  of 
railway  curves  ;  table  of  degrees  of  curva- 
ture, radii,  and  central  ordinates  ;  railway 
plot  and  profile  ;  the  two  combined ;  pro- 
file and  cross-section  paper ;  regulations 
of  the  English  Parliament  for  railway 
plans  ;  geographical  sections  ;  hydro- 
graphic  and  marine  surveys  ;  rough  draft ; 
transferring ;  tracing ;  photography ;  copy- 
ing glass ;  transfer  paper ;  reduction  and 
enlargement  of  plans,  355-369. 

Finishing  plan  ;  direction  of  light ;  boundary 
lilies  ;  lettering ;  examples  of  alphabets ; 
construction  of  letters  mechanically ; 
spacing  of  letters;  lines  of  lettering; 
titles,  369-377. 

Tinted  topographical  drawing ;  conventional 
tints ;  colors  used  by  French  military  en- 
gineers ;  imitation  of  conventional  signs ; 
representations  of  hills,  woods,  rivers,  by 
the  brush  ;  effect  of  oblique  light ;  prep- 
aration of  paper  for  tinted  drawing ; 
application  of  tints,  lettering,  flourishes, 
378-382. 

Map  of  portion  of  the  city  of  London,  show- 
ing drainings,  contours,  gas  and  water 
mains,  and  occupancy  of  buildings;  a 
larger  portion,  showing  effect  of  contour 
lines,  382-284. 

PERSPECTIVE  DRAWING. — Direction  of  lumin- 
ous rays ;  angle  at  which  objects  can  be 
seen;  linear  and  aerial  perspective;  the 
planes  of  a  picture  ;  point  of  sight ;  par- 
allel and  angular  perspective  ;  to  draw  a 
square  and  cube  in  parallel  perspective ; 
construction  of  a  scale ;  to  determine  the 
position  of  any  point  in  the  ground  plane ; 
to  draw  an  octagon,  a  circle,  a  pyramid, 
a  cone,  in  parallel  perspective  ;  to  draw  a 
square,  cube,  octagonal  pillar,  circular 
pillar,  octagonal  pyramid,  cone,  elevation 
of  a  building,  arched  bridge,  in  angular 
perspective;  to  draw  the  interior  of  a 
room,  a  flight  of  stairs,  and  to  find  the 


viii 


TABLE   OF   CONTENTS. 


reflections  of  objects  in  water;  the  per- 
spective projection  of  shadows ;  the  most 
agreeable  angle  of  vision  for  a  represen- 
tation in  perspective,  385-404. 

ISOMETRICAL  DsAwiNG. — Principle  of  iso- 
metrical  representation ;  projection  of  a 
cube  and  its  general  application ;  collec- 
tion of  cubes  and  sections  of  cubes ;  pro- 
jection of  curved  lines;  division  of  the 
circumference  of  a  circle  ;  projection  of  a 
bevel  wheel,  of  a  pillar  block,  of  a  cul- 
vert, section  of  a  boiler,  bridge  truss; 
horizontal  section  or  plan  of  school-house ; 
portion  of  a  roof  truss ;  perspective  draw- 
ings, in  which  the  point  of  sight  is  above 
the  plane  of  the  picture,  415-424. 

ENGINEERING  DRAWING. — Tredgold's  defini- 
tion of  civil  engineering;  foundations; 
piles ;  rule  to  find  the  weight  .which  a 
pile  iwill  bear ;  weight  of  ram ;  size  of 
piles ;  sheet  piling ;  hollow  cast-iron  piles, 
how  driven ;  Harlem  bridge ;  coffer  dams ; 
foundation  of  Susquehanna  bridge ;  sec- 
tion of  river-wall,  Thames  embankment, 
extracts  from  specifications  for ;  crib  pier 
for  quarantine  establishment  for  the  port 
of  New  York,  extracts  from  specifications 
for  same,  415-424. 

Dams  across  the  Connecticut  River  at  Hoi- 
yoke,  Mass. ;  across  Merrimack,  at  Lowell, 
Mass. ;  across  Mohawk,  at  Cohoes,  K  Y. ; 
across  Croton  Eiver,  N.  Y.  ;  gauging 
of  streams;  rain-fall  and  evaporation; 
head-gates  at  Cohoes  Dam ;  hoisting  ap- 
paratus for  same,  424-431. 

Canals,  dimensions  of;  section  of  Erie;  of 
Northern,  at  Lowell ;  wall  of  same ;  locks 
of  canals ;  details  of  Chemung  and  Erie, 
enlarged ;  extracts  from  specifications  of 
New  York  State  canals ;  ponds  or  reser- 
voirs ;  flumes ;  headgates  of  flume  at  Hoi- 
yoke;  section  of  conduit  of  Brooklyn 
City  Water-works ;  of  Croton  Aqueduct ; 
pipes  across  Harlem  Eiver ;  Croton  new 
reservoir :  extracts  from  specification  for 
same  ;  water  mains ;  dimension  of  Brook- 
lyn Water- works  pipes ;  formulas  for 


discharge  of  pipes ;  extracts  from  speci- 
fication for  Brooklyn  pipes,  weights  of; 
sewers,  dimensions  of ;  man-holes;  catch- 
basins,  431-448. 

Gas  supply ;  weight  of  pipes  ;  discharge  of, 
449. 

Roads  and  streets;  division  and  width  of, 
here  and  in  Paris ;  pave  ;  carriage-way ; 
Belgian,  wooden,  and  asphalt  pave  ;  roads, 
McAdam,  Telford,  Central  Park;  grades 
of;  table  of  inclinations,  feet  per  mile, 
and  angles;  railways;  gauge,  width  of 
cut  and  embankment ;  resistance  of  car- 
riages on  roads,  450-455. 

Bridges ;  piers ;  trestles ;  arches,  table  of 
dimensions  of  several ;  rule  for  depth  of 
key ;  height  of  spandrel ;  thrust  of  arch  ; 
resistance  of  abutment  ;  skew  arch  ; 
frame  bridge ;  tension  and  compression  of 
chords ;  specification  for  Howe's  truss ; 
Cubitt's  cast-iron  girders;  wrought-iron 
truss  across  Connecticut  River ;  suspen- 
sion bridges,  dimensions  of  several,  455- 
466. 

Steam  engines  ;  horse-power ;  boilers,  evap- 
oration of;  rate  of  combustion ;  work- 
ing-pressure ;  strength  of  shell  and  flues ; 
joints,  stays;  locomotive  fire-box;  chim- 
neys ;  size  of  flues  ;  stationary  engine, 
details  of,  stuffing-box,  piston ;  founda- 
tion for  engine,  466-474. 

PROJECTIONS  OF  THE  GLOBE. — Globular  and 
stereographic  projections  of  hemisphere  ; 
construction  of  maps  by  development ; 
table  of  miles  to  degrees  of  longitude ; 
Mercator's  chart,  construction  of,  475- 
481. 

!  Specifications,  form  of,  482. 
j  APPENDIX. — Extracts    from    Building  Act, 
'city  of  New  York,  483-485. 

Francis's  tables  on  size  of  shafting ;  power 
required    to    drive    cotton  and   woollen 
'  mill,  486-488. 

Profile  and  cross-section  paper  ;  application 
of  profiles  to  flow  of  water,  resistance 
and  movement  of  trains,  489,  490. 

INDEX,  491-496. 


LIST    OF    PLATES. 


PAGE 

PLATES  I.,  II.     Projections  of  a  regular  Hexagonal  Pyramid,        ....  86 

PLATES  III.,  IV.     Projections  of  Prisms, 88 

PLATE  V.     Conic  Sections, 90 

PLATES  VI.,  VII.     Penetrations  of  Cylinders, 92 

PLATES  VIII.,  IX.     Penetrations  of  Cylinders,  Cones,  and  Spheres,       ...  94 

PLATES  X.,  XI.     Penetrations  of  Cylinders,  Prisms,  Spheres,  and  Cones,          .  98 

PLATE  XII.     The  Spiral  and  Helix, 100 

PLATE  XIII.     Development  of  the  Surface  of  Intersected  Cylinders  and  Cones,  102 

DRAWING   OF  MACHINERY. 

PLATE  XIV.     Drawings  of  Water-wheel  Shafts,     ....  139 

PLATE  XV.             "         of  a  Standard, 146 

PLATE  XVI.            "         of  a  Sprawl  and  bracket  Hanger,     ......  148 

PLATES  XVII.,  XVIII.     Drawings  of  Spur  Wheels, 176 

PLATES  XIX.,  XX.     Oblique  Projections  of  a  Spur  Wheel, 178 

PLATES  XXI.,  XXII.     Projections  of  a  Bevel  Wheel, 180 

PLATES  XXIII.,  XXIV.     Rack  Gear  and  Pinion,  Worm  and  Wheel,      .        .        .183 

PLATES  XXV.,  XXVI.     Internal  Gearing, 184 

PLATE  XXVII.     Projections  of  Eccentrics, 185 

DRAWING   OF   SCREWS. 

PLATES  XXVIH.,  XXIX.    Projections  of  a  Triangular-threaded  Screw  and  Nut,  189 

FRAMES. 

PL  ATE  XXX.     Iron  Frames  of  Tools, 192 

PLATE  XXXI.     Elevation  of  the  Frames  of  American  Marine  Engines,    .        .  194 

PLATE  XXXII.     Working-Beams  and  Cranks, 195 

PLATE  XXXIII.     Steam-engine  Connecting  Rods,  and  Details,         .         .         .  197 


LIST   OF    PLATES. 


LOCATION   OP   MACHINES. 

PAGE 

PLATES  XXXIV.,  XXXV.    Plans  of  the  Location  of  Machines,    ....  200 
PLATES  XXXVI.,  XXXVII.,  XXXVIII.     Elevation,  Section,  and  Plan   of  the 

48"  Stop-gate  in  use  at  the  Nassau  Water-works,  Brooklyn,  L.  I.,  .        .  202 

PLATE  XXXIX.     Sections  of  a  Locomotive  Boiler, 204 

PLATE  XL.    Front  Elevation  of  one  of  the  Engines  of  the  Golden  Gate,        .  204 

PLATE  XLI.     Side  Elevation  of  the  same, 204 

PLATE  XLLT.     Vertical  Section  through  the  Centre  of  a  Turbine  Wheel,  .        .  207 

PLATE  XLIII.    Plan  of  the  Turbine  Wheel  and  Wheel-pit, 207 

PLATE  XLIV.    Plan  of  the  Wheel,  Guides,  and  Garniture,       ....  208 


ARCHITECTURAL    DRAWING. 

PLATE  XLV.    Elevations  and  Details  of  Framed  Roofs, 220 

PLATE  XL VI.    Iron  Roof  and  Trussed  Girders, 231 

PLATES  XLVII.-LI.     Plans  and  Elevations  of  a  House, 248 

PLATE  LII.     Example  of  the  Tuscan  Order, 254 

PLATE  LILT.          "          of  the  Doric  Order,    .        . 254 

PLATE  LIV.          "          of  the  Ionic  Order,        .                 255 

PLATE  LV.           "        of  the  Corinthian  Order, 256 

PLATE  LVI.     Roman  Arches  and  Entablatures,  Gothic  and  Byzantine  Columns,  260 

PLATE  LVII.     Buttresses,  Towers,  and  Spires, .268 

PLATE  LVIII.     Windows, 270 

PLATE  LIX.     Doorways, 271 

PLATE  LX.     Ornaments  and  Brackets, 274 

PLATE  LXI.     Roman  and  Saracenic  Ornament, 274 

PLATE  LXII.     Ornamental  Mouldings, 276 

PLATE  LXIII.     Ornaments  of  the  Renaissance, 278 

PLATE  LXIV.    Front  Elevation  of  a  High-Stoop  House,  New  York  City,        .  280 
PLATE  LXV.    Elevation  of  a  House,  from  "  Holly's  Country  Seats,"    .        .        .282 

PLATE  LXVI.    Plan  and  Elevation  of  a  Farm-house,  in  the  English  Rural  Style,  281 
PLATE  LXVII.    Elevation  and  Plan  of  a  plain  Timber  Cottage  Villa,  .        .        .281 

PLATE  LXVIII.     A  Villa,  Rural  Gothic  Style, 282 

PLATE  LXIX.     A  Villa  in  the  Italian  Style, 284 

PLATE  LXX.     A  Tenement  House, 284 

PLATE  LXXI.     Elevation  of  a  Store  Front, 286 

PLATE  LXXII.  Fa9ade  of  two  Stores  erected  on  Broadway  ....  285 
PLATE  LXXIII.  Elevation  of  Store  Front,  executed  in  Cast  Iron,  .  .  .  288 
PLATE  LXXIV.  Plan  and  Elevation  of  a  District  School-house,  ...  288 
PLATE  LXXV.  Design  for  a  Church  in  the  English  Decorated  Gothic  Style,  .  294 
PLATE  LXXVI.  Front  Elevation  of  the  Roman  Catholic  Cathedral,  Fifth  Ave- 
nue, New  York  City, 294 

PLATE  LXXVII.     Byzantine  Church,  Park  Avenue, 295 

PLATE  LXXVIII.    Interior  Perspective  View  of  the  New  York  Crystal  Palace,  297 


LIST    OF    PLATES. 


SHADING   AND    SHADOWS. 

PAGE 

PLATE  LXXIX.     Forms  of  Shadows, 314 

PLATE  LXXX.     Outlines  of  Shadows  cast  upon  two  Planes  of  Projection,      .  318 
PLATE  LXXXI.     Outlines  of  Shadows  cast  upon  the  Interior  of  a  hollow  Cyl- 
inder and  Ring,    ............  325 

PLATE  LXXXII.     Outlines  of  Shadows  cast  upon  the  Surfaces  of  Screws  and 

Nuts,  both  Triangular  and  Square-threaded, 327 

METHODS   OF   TINTING. 

PLATE  LXXXIII.     Shading  by  Flat  Tints,     ......        K        ..  330 

PLATE  LXXXIV.     Shading  by  Softened  Tints,          ......  332 

ELABORATION  OF  SHADING  AND  SHADOWS. 

PLATES  LXXXV.,  LXXXVI.     Examples  in  Lithography  of  Shades  and  Shadows 

of  different  Solids, 336 

PLATE  LXXXVII.     Effects  of  Light,  Shade,  and  Shadow,  on  Screws,      .        .  338 

FINISHED    COLORING. 

PLATES  LXXXVIII.  and  LXXXIX.     Illustrations  in  Chromo-Lithography  from 

Colored  Drawings, 348 

TOPOGRAPHICAL   DRAWING. 

PLATE  XC.     Examples  of  Topographical  Drawings, 349 

PLATE  XCI.    Meridians  and  Borders, 370 

PLATE  XCII.     Mechanical  Method  of  Constructing  Letters  and  Figures,          .  374 
PLATE  XCIII.     Example  of  Titles  illustrating  the  Form  of  Letters,      .        .        .374 

PLATE  XCIV.    Map  of  the  Harbor  and  City  of  New  Haven,     ....  382 

PLATE  XCV.     Examples  of  topographical  Drawing, 382 

PLATE  XCVI.          "          "              "                    "          in  colors,        ...  382 

PLATE  XCVII.     Geological  Map,  from  Blake's  "  Survey  of  California,"        .        .  382 

FRONTISPIECE.  r 

PLATE  XCVHI.     Photograph  from  a  Drawing  of  the  Engine  and  Boiler  of  the 

Steamer  Pacific, 388 

PERSPECTIVE    DRAWING. 

PLATE  XCIX.    A  Square  and  Cube  in  Parallel  Perspective, 390 

PLATE  C.                     «            »          in  Angular  Perspective,      ....  396 
PLATE  CI.    Projection  of  an  Octagonal  Pillar,  Cylinder,  Pyramid,  and  Cone,  in 

Angular  Perspective, 398 

PLATE  CII.     The  Elevation  of  a  Building  in  Angular  Perspective,    .        .        .  400 

PLATE  CHI.     An  Arched  Bridge  in  Angular  Perspective,  and  Interior  of  a  Room,  400 


xii  LIST   OF   PLATES. 

PAGE 

PLATE  CIV.    A  Flight  of  Stairs,  and  Reflections  in  Water,      ....  402 

PLATE  CV.    Perspective  Projection  of  Shadows, 402 

ISOMETRICAL    DRAWING. 

PLATE  CVI.  Sections  of  Cubes, 408 

PLATE  CVII.  Bevel  Wheel  and  Pillow  Block, 409 

PLATE  CVIII.  Projection  of  a  Culvert,  such  as  were  built  beneath  Croton 

Aqueduct, 409 

PLATE  CIX.  Elevation  and  Section  in  Isometry  of  the  District  School-house 

given  in  Plate  LXXIX., 412 

ENGINEERING   DRAWING. 

PLATE  CX.  Transverse  Section  of  the  River-wall,  Thames  Embankment,  .  421 
PLATE  CXI.  Isometrical  View  of  the  Overflow  and  Outlet  of  the  Victoria  and 

Regent  Street  Sewers  in  the  Thames  Embankment, 423 

PLATE  CXII.  Section  of  the  Dam  across  the  Connecticut  River,  at  Holyokc, 

Mass., 430 

PLATES  CXHI.  and  CXIV.  Drawings  in  Plan  and  Detail  of  the  Headgates,  and 

the  Machinery  for  hoisting  them,  at  the  Cohoes  Company  Dam,  .  .  430 

PLATE  CXV.  Elevation  and  Section  of  the  Headgates  of  a  Flume,  .  .  438 

PLATE  CXVI.  Sections  of  the  Fire-box  of  a  Locomotive, 470 

PLATE  CXVII.  Elevations  and  Section  Tubular  Boiler,  .  ...  471 

PLATE  CXVIII.  Sections  of  Chimneys, 472 

PLATE  CXIX.  Elevation  and  Details  of  a  Stationary  Engine,  .  .  .  473 

PLATE  CXX.  Plan  and  Elevations  of  Foundation  of  a  Stationary  Engine,  .  474 


CYCLOPAEDIA    OF    DRAWING. 


GEOMETRICAL  DEFINITIONS  AND  TECHNICALITIES. 

A  point  is  mere  position  without  magnitude,  as  the  intersection  of  two 
lines,  or  the  centre  of  a  circle. 

Lines  are  measured  by  length  merely,  and  may  be  straight  or  curved. 
Straight  lines  are  generally  designated  by  letters  or  figures  at  their  ex- 
tremities, as  the  line  A —  — B,  the  line  1 —  — 2.  Curved  lines,  by 
additional  intermediate  letters  or  figures,  as  the  curved  line  ABC. 

A  given  point  or  given  line  expresses  a  point  or  line  of  fixed  position 
or  dimension. 

Surfaces  or  superficies  are  measured  by  length  and  breadth  only.  They 
may  be  plane  or  curved. 

Solids  are  measured  by  length,  breadth,  and  thickness.  The  extremi- 
ties of  lines  are  points,  the  boundaries  of  surfaces  are  lines,  and  the  boun- 
daries of  solids  are  surfaces.  

Parallel  lines  are  lines  in  the  same  plane 
which  are  equally  distant  from  each  other    

Fig.  1. 

at  every  part  (fig.  1). 

Horizontal  lines  are  such  as  are  parallel  to  the  horizon,  or  level. 

Vertical  lines  are  such  as  are  parallel  to  the  position  of  a  plumb-line 
suspended  freely  in  a  still  atmosphere. 

Inclined  lines  occupy  an  intermediate  between  horizontal  and  vertical 
lines.    Also  two  lines  which  converge  towards  each  other,  and  if  produced, 
would  meet  or  intersect,  are  said  to  incline  to  each  other. 
1 


GEOMETRICAL   DEFINITIONS   AND   TECHNICALITIES. 


An  angle  is  the  opening  between  two  straight  lines  which  meet  one 
another.  "When  several  angles  are  at 
one  point  B,  any  one  of  them  is  expressed 
by  three  letters,  of  which  the  letter  that 
is  at  the  vertex  of  the  angle,  that  is,  at 
the  point  in  which  the  straight  lines  that 
contain  the  angle  meet  one  another,  is 
put  between  the  other  two  letters  :  Thus 
the  angle  which  is  contained  by  the 

straight  lines,  AB,  CB,  is  named  the  angle  ABC,  or  CBA ;  but  if  there  be 
only  one  angle  at  a  point,  it  may  be  expressed  by  a  letter  placed  at  that 
point ;  as  the  angle  at  E.' 

When  a  straight  line  standing  on  another  straight  line  makes  the  adja- 
cent angles  equal  to  one  another,  each  of  the  angles  is  called  a  right  angle  / 
and  the  straight  lines  are  said  to  be  perpendicular  to  each  other  (fig.  3). 
An  obtuse  angle  is  that  which  is  greater  than  a  right  angle  (fig.  4). 


Fig.  3. 


Fig.  4 


Fig.  5. 


Fig.  6. 


An  acute  angle  is  that  which  is  less  than  a  right  angle  (fig.  5). 

A  triangle  is  a  flat  surface  bounded  by  three  straight  lines ;  when  the 
three  sides  are  equal,  the  triangle  is  equilateral ;  when  only  two  of  its 
sides  are  equal,  isosceles;  when  none  equal,  scaline;  when  one  of  the 
angles  is  a  right  angle,  the  triangle  is  right  angled,  and  then  the  longest 
side,  or  that  opposite  the  right  angle  is  called  the  hypothenuse.  The 
upper  extremity  of  the  triangle  is  called  the  apex,  the  bottom  line  the 
lose,  and  the  two  other  including  lines  the  sides. 

A  Quadrilateral  figure  is  a  surface  bounded  by  four  straight   lines. 


Fig.  7.  Fig.  8.  Fig.  9.  Fig.  10.  . 

When  the  opposite  sides  are  parallel,  it  is  a  parallelogram ;  if  its  angles 


GEOMETRICAL   DEFINITIONS   AND   TECHNICALITIES. 


8 


are  right  angles,  it  is  a  rectangle  (fig.  7) ;  if  the  sides  are  also  equal,  it  is  a 
square  (fig.  8) ;  if  all  the  sides  are  equal,  but  the  angles  not  right  angles, 
it  is  a  rhombus  (fig.  9).  A  trapezium  has  only  two  of  its  sides  parallel  (fig. 
10).  A  diagonal  is  a  straight  line  joining  two  opposite  angles  of  a  figure. 
Plane  figures  of  more  than  four  sides  are  called  polygons.  When  the 

Fig.  12.  Fig.  13. 


Fig.  11. 


Fig.  14. 


Fig.  15. 


Pentagon,  five  sides.  Hexagon,  six  sides.  Heptagon,  seven  sides.  Octagon,  eight  sides. 

sides  are  equal,  they  are  regular  polygons ;  of  which  figs.  11-14  are  ex- 
amples, annexed  to  which  are  their  respective  designations. 

A  circle  is  a  plane  figure  contained  by  one  line, 
which  is  called  the  circumference,  and  is  such  that  all 
straight  lines,  drawn  from  a  certain  point  within  the 
figure  to  the  circumference,  are  equal  to  one  another. 
And  this  point  is  called  the  centre  of  the  circle. 

The  term  circle  is  very  generally  used  for  the  cir- 
cumference, and  will  be  found  to  be  employed  in  this 
work  with  this  twofold  meaning. 

Any  straight  line  drawn  from  the  centre  and  terminating  in  the  cir- 
cumference is  termed  a  radius  /  if  drawn  through  the  centre,  and  termi- 
nated at  each  end  by  the  circumference,  it  is  termed  a  diameter. 

An  arc  of  a  circle  is  any  part  of  the  circumference. 

A  sector  of  a  circle  is  the  space  enclosed  by  two  radii  and  the  inter- 
cepted arc.  When  the  radii  are  at  right  angles,  the  space  is  called  a  quad- 
rant or  one-fourth  of  a  circle.  Half  a  circle  is  called  a  semicircle. 

A  chord  is  a  straight  line  joining  -the 
extremities  of  an  arc,  as  a  1).  The  space 
cut  off'  by  the  chord  is  termed  a  segment. 

A  tangent  to  a  circle  or  other  curve 
is  a  straight  line  which  touches  it  at  only 
one  point,  as  c  d  touching  the  circle  at 
only  e. 

Circles  are  concentric  when  described  Fig.  is. 

from  the  same  centres.     Eccentric  when  described  from  different  centres. 

Triangular  or  other  figures  with  a  greater  number  of  sides  are  inscribed 


GEOMETRICAL   DEFINITIONS   AND   TECHNICALITIES. 


rig.  17. 


Fig.  18. 


in,  or  circumscribed  ~by  a  circle,  when  the  vertices  of  all  their  angles  are  in 

the  circumference  (fig.  17). 

A  circle  is  inscribed  in  a 
straight-sided  figure,  when  it 
is  tangent  to  all  the  sides  (fig. 
18). 

All  regular  polygons  may 
be  inscribed  in   circles,  and 
circles  may  be   inscribed  in 
polygons;  hence  the  facility 
With  which  polygons  may  be  constructed. 

For  the  measurement  of  angles,  the  circumference  of  a  circle  is  divided 

into  360  equal  arcs,  called  degrees  °, 
which  are  again  subdivided  into  min- 
utes '  and  seconds  " ;  60  minutes  to  a 
degree,  and  60  seconds  to  a  minute, 
the  vertex  of  the  angle  being  placed 
at  the  centre  of  the  circle,  the  angle 
is  measured  by  the  arc  enclosed  be^- 
tween  the  sides.  Thus  the  angle 
DCB  is  measured  by  the  arc  DB ;  the 
line  DH,  a  line  drawn  from  one  ex- 
tremity of  the  arc  perpendicular  to 
the  radius  passing  through  the  other 
extremity  is  called  the  sine  of  the 
angle,  GD  is  the  cosine,  HB  the  versed 
sine,  AB  the  tangent,  FE  the  cotangent,  AC  the  secant,  and  CE  the  cosecant. 
An  ellipse  is  an  oval-shaped  curve  from  any  point  P  in  which,  if  straight 

lines  be  drawn  to  two  fixed  points 
FF;,  their  sum  will  be  always  the 
same.  FF'  are  the  foci,  the  line 
passing  through  the  foci  is 'called 
the  transverse  axis,  the  line  CD 
\S  perpendicular  to  the  centre  of 
this  line  the  conjugate  axis. 

A  parabola  is  a  curve  in 
which  any  point  P  is  equally  dis- 
tant from  a  certain  fixed  point 
F  and  a  straight  line  KK' ;  thus, 


GEOMETRICAL,   DEFINITIONS   AND   TECHNICALITIES.  5 

PF.  is  always  equal  to  PD=.    F  is  called  the  focus,  and  the  line  KK'  the 
directrix  (fig.  21). 
K 


Fig.  21. 


Fig.  22. 


An  hyperbola  is  a  curve  from  any  point  P  in  which,  if  two  straight  lines 
be  drawn  to  two  fixed  points  FF'  the  foci,  their  difference  shall  always  be 
the  same  (fig.  22). 

A  cycloid  is  the  curve  described  by  a  point  P  in  the  circumference  of 


Fig.  23. 


Fig.  24. 


a  circle  which  rolls  along  an  extended  straight  line  until  it  has  completed 
a  revolution. 

If  the  circle  be  rolled  on  the  circumference  of  another  circle,  the  curve 
then  described  by  the  point  P  is  called  an  epicycloid  (fig.  24). 

Epicycloids  are  external  or  internal,  according  as  the  rolling  or  gener- 
ating circle  revolves  on  the  outside  or  inside  of  the  fundamental  circle. 
The  internal  epicycloid  is  sometimes  called  a  hypocycloid. 


OF   SOLIDS. 


A  prism  is  a  solid  of  which  the  ends  are  equal,  similar,  and  parallel 
straight-sided  figures,  and  of  which  the  other  sides  are  parallelograms. 
When  all  the  sides  are  squares,  it  is  called  a  cube  (fig.  25). 

A  pyramid  is  a  solid  having  a  straight-sided  base,  and  triangular  sides 
terminating  in  one  point  or  vertex  (fig.  26). 

Prisms  and  pyramids  are  distinguished  as  triangular,  quadrangular, 


GEOMETRICAL   DEFINITIONS  AND   TECHNICALITIES. 


pentagonal,  hexagonal,  &c.,  according  as  the  base  has  three,  four,  five,  six 
sides,  <fec. 


Fig.  25. 


Fig.  26. 


Fig.  27. 


A  sphere  or  globe  (fig.  27),  is  a  solid  bounded  by  a  uniformly  curved 
surface,  every  point  of  which  is  equally  distant  from  the  centre,  a  point 
within  the  sphere.  A  line  passing  through  the  centre,  and  terminating 
both  ways  at  the  surface,  is  a  diameter. 


Fig.  29.  Fig.  30. 

A  cylinder  is  a  round  solid  of  uniform  thickness,  of  which  the  ends  are 
equal  and  parallel  circles  (fig.  28). 

A  cone  is  a  round  solid,  with  a  circle  for  its  base,  and  tapering  uni- 
formly to  a  point  at  the  top  (fig.  29). 

When  a  solid  is  cut  through  transversely  by  a  plane  parallel  to  the 
base,  the  part  cut  off  is  a  segment,  and  the  part  remaining  is  afrustrum  ol 
the  solid.  The  latter  term  is  usually  limited  to  pyramids  and  cones. 


Fig.  81. 


Fig.  82. 


Fig.  33. 


Fig.  34. 


The  tetrahedron,  bounded  by  four  equilateral  triangles  (fig.  30). 
The  fexahedron,  or  cube,  bounded  by  six  squares  (fig.  31). 
The  octahedron,  bounded  by  eight  equilateral  triangles  (fig.  32). 
The  dodecahedron,  bounded  by  twelve  pentagons  (fig.  33). 
The  icosahedron,  bounded  by  twenty  equilateral  triangles  (fig.  34). 
Regular  solids  may  be  circumscribed  by  spheres,  and  spheres  may  bo 
inscribed  in  regular  solids. 


DRAWING  INSTRUMENTS. 


DRAWING   INSTRUMENTS. 

Lead  pencil. — Pencils  are  of  various  qualities,  distinguished  by  letter 
marks,  of  which  the  most  common  in  use  by  draftsmen  are  HH  and  HUH. 
The  pencil  used  for  drawing  straight  lines  should  be  sharpened  to  a  chisel 
edge ;  for  making  dots  and  marking  points  (.),  the  pencil  should  have  a 
round  sharp  point.  Pencil  lines  intended  to  be  made  permanent  in  ink, 
should  be  drawn  quite  delicately.  The  pencil  should  not  be  held  tightly ; 
a  slight  hold  without  slackness,  inclined  a  little  to  the  side  toward  which 
the  line  is  drawn.  Never  extend  the  line  beyond  what  is  necessary,  and 
avoid  as  much  as  possible  the  use  of  rubber,  as  it  roughs  the  paper,  mak- 
ing it  difficult  to  trace  a  smooth  line  in  ink,  and  readier  to  receive  and 
retain  dust. 

The  common  ruler  or  straight  edge. — Rulers  should  be  of  close-grained, 
thoroughly  seasoned  wood,  such  as  mahogany,  maple,  pear,  &c.  They 
should  be  about  \  of  an  inch  thick,  bevelled  a  little  on  one  edge,  and  from 
1  to  2i  inches  wide,  according  to  their  length.  Every  draftsman  should 
have  at  least  two  rulers,  the  shortest  from  9  inches  to  a  foot  long,  and  the 
other  as  long  as  he  may  require  in  his  drawing.  As  the  accuracy  of  a 
drawing  depends  greatly  on  the  straightness  of  the  lines,  the  bevelled  edge 
of  the  ruler  should  be  perfectly  straight.  To  test  this,  place  a  sheet  of 
paper  on  a  perfectly  smooth  board ;  insert  two  very  fine  needles  in  an  up- 
right position  through  the  paper  into  the  board,  distant  from  each  other 
nearly  the  length  of  the  ruler  to  be  tested ;  bring  the  edge  of  the  ruler 
against  these  needles,  and  draw  a  line  from  one  needle  to  the  other ;  re- 
verse the  ruler,  bringing  the  same  edge  on  the  opposite  side  and  against 
the  needles,  and  again  draw  a  line.  If  the  two  lines  coincide,  the  edge  is 
straight ;  but  if  they  disagree,  the  ruler  is  inaccurate.  When  one  ruler 
has  been  tested,  the  other  can  be  examined  by  placing  their  edges  against 
the  correct  one,  and  holding  them  between  the  eye  and  the  light. 

Triangles  are  made  of  the  same  kinds  of  wood  as  the  ruler,  and  some- 
what thinner,  and  of  various  sizes.  They  should  be  right-angled,  with  acute 
angles  of  45°,  or  of  60°  and  30°.  The  most  convenient  size  for  general  use 
measures  from  3  to  6  inches  on  the  side.  A  larger  size  from  8  to  10  inches 


DRAWING   INSTRUMENTS. 


o 


Fig.  85. 


long  on  the  side  is  convenient  for  making,  drawings  to  a  large  scale.  Cir- 
cular openings  are  made  in  the  body  of  the  triangle  for  the  insertion  of  the 
end  of  the  finger  to  give  facility  in  sliding  the  triangle  on  the  paper.  Tri- 
angles are  sometimes  made  as  large  as  15  to  18  inches  on  the  side  ;  but  in 
this  case  they  are  framed  in  three  pieces  of  about  1£ 
wide,  leaving  the  centre  of  the  triangle  open.  The 
value  of  the  triangle  in  drawing  perpendicular 
lines  depends  on  the  accuracy  of  the  right  angle. 
To  test  this  (fig.  36),  draw  a  line  with  an  accurate 
ruler  on  paper.  Place  the  right  angle  of  the  tri- 
angle near  the  centre  of  this  line,  and  make  one 
of  the  adjacent  sides  to  coincide  with  the  line ;  now 
draw  a  line  along  the  other,  adjacent  side,  which, 
if  the  angle  is  strictly  a  right  angle,  will  be  per- 
pendicular to  the  first  line.  Turn  the  triangle  on 
this  perpendicular  side,  bringing  it  into 
the  position  ABC7 ;  if  now  the  sides  of 
the  triangle  agree  with  the  line  BC'  and 
AB,  the  angle  is  a  right  angle,  and  the 
sides  straight.  The  straightness  of  the 
hypothenuse  or  longest  side  can  be  tested 
like  a  common  ruler. 
The  triangle  is  used  for  the  drawing  of  lines  parallel  or  perpendicular 
to  each  other.  Thus  (fig.  37),  if  it  were  required  to  draw  lines  parallel 

and  perpendicular  to  c  d,  place 
one  side  of  the  triangle  so  as  to 
coincide  accurately  with  the  given 
linec<?/  keeping  the  triangle  in 
this  position  with  the  right  hand, 
bring  the  edge  of  the  ruler  against 
the  hypothenuse  of  the  triangle ;  if 
now  the  ruler  be  held  securely  by 
the  left  hand,  the  triangle  may  be 
slid  along  the  edge  of  the  ruler, 

and  any  line  drawn  along  the  upper  side  of  the  triangle  will  be  parallel  to 
the  line  c  d,  and  the  lines  drawn  along  the  other  side  of  the  triangle  will 
be  perpendicular  to  this  same  line ;  in  this  way  a  rectangle  may  be  drawn 
through  three  given  points  without  moving  the  position  of  the  ruler. 

It  is  evident  that  for  the  drawing  of  parallel  lines  merely,  either  side 
may  be  brought  in  contact  with  the  ruler ;  but  the  longer  the  side  in  con- 


S 

Fig.  86. 


DRAWING    INSTRUMENTS.  9 

tact,  the  more  accurately  may  the  parallelism  be  preserved  in  sliding  the 
triangle. 

The  T  square  is  a  thin  "straight  edge"  or  ruler,  «,  fitted  at  one 
end  with  a  stock,  5,  applied  transversely  at  right  angles.  The  stock 
being  so  formed  as  to  fit  and  slide  against  one  edge  of  the  drawhig  board, 
the  blade  reaches  over  the  surface,  and  presents  an  edge  of  its  own  at  right 
angles  to  that  of  the 
board,  by  which  par- 
allel straight  lines 
may  be  drawn  upon 
the  paper.  To  suit 


a  41-inch  board,  the 
blade  should  meas-  Fis- 88- 

ure  40  inches  long  clear  of  the  stock,  or  one  inch  shorter  than  the  board, 
to  remove  risk  of  injury  by  overhanging  at  the  end ;  it  should  be  2£ 
inches  broad  by  -^  inch  thick,  as  this  section  makes  it  sufficiently  stiff 
laterally  and  vertically.  The  tip  of  the  blade  may  be  secured  from 
splitting  by  binding  it  with  a  thin  strip  inserted  in  a  saw-cut.  The 
stock  should  be  14  inches  long,  to  give  sufficient  bearing  on  the  edge  of 
the  board,  2  inches  broad  and  £  inch  thick,  in  two  equal  thicknesses 
glued  together.  With  a  blade  and  stock  of  these  sizes,  a  well  propor- 
tioned T  square  may  be  made,  and  the  stock  will  be  heavy  enough  to  act 
as  a  balance  to  the  blade,  and  to  relieve  the  operation  of  handling  the 
square.  The  blade  should  be  sunk  flush  into  the  upper  half  of  the  stock 
on  the  inside,  and  very  exactly  fitted.  It  should  be  inserted  full  breadth, 
as  shown  in  the  figure  ;  notching  and  dovetailing  is  a  mistake,  as  it  weak- 
ens the  blade,  and  adds  nothing  to  the  security.  The  lower  half  of  the 
stock  should  be  only  If  inches  broad,  to  leave  a  |-inch  check  or  lap,  by 
which  the  upper  half  rests  firmly  on  the  board  and  secures  the  blade  lying 
flatly  on  the  paper. 

For  the  smaller  sizes  of  board,  reduce  the  proportions  of  both  the  blade 
and  the  stock. 


Fig.  39 

One  half  of  the  stock,  c  (fig.  39),  is  in  some  cases  made  loose,  to  turn 
upon  a  brass  swivel  to  any  angle  with  the  blade  a,  and  to  be  clenched  by 


10 


DRAWING   INSTRUMENTS. 


a  screAved  nut  and  washer.  The  loose  stock  is  useful  for  drawing  parallel 
lines  obliquely  to  the  edges  of  the  board,  such  as  the  threads  of  screws, 
oblique  columns,  and  connecting-rods  of  steam-engines.  A  square  of  this 
sort  should  be  rather  as  an  addition  to  the  fixed  square,  and  used  only 
when  the  "bevel  edge  is  required,  as  it  is  not  so  handy  as  the  other. 

The  edges  of  the  blade  should  be  very  slightly  rounded,  as  the  pen  will 
thereby  work  the  more  freely.  It  is  a  mistake  to  chamfer  the  edges,  that 
is,  to  plane  them  down  to  a  very  thin  edge,  as  is  sometimes  done  with  the 
object  of  insuring  a  correct  position  of  the  lines ;  for  the  edge  is  easily 
damaged,  and  the  pen  is  liable  to  catch  the  edge,  and  to  leave  ink  upon  it. 
A  small  hole  should  be  made  in  the  blade  near  the  end,  by  which  the 
square  may  be  hung  up. 

In  many  drawing  cases  will  be  found  the  parallel  ruler  (fig.  40),  consist- 
ing of  two  rulers  connected 

by   two   bars    moving   on 

1        pivots,  and  so  adjusted  that 

the  rulers,  as  they  open, 
form  the  sides  of  a  paral- 
lelogram. The  edge  of  one 
of  the  rulers  being  retained 
in  a  position  coinciding 

with,  or  parallel  to  a  given  line  ;  the  other  ruler  may  be  moved,  and  lines 
drawn  along  its  edge  must  also  be  parallel  to  the  given  line.  This  instru- 
ment is  only  useful  in  drawing  small  parallels,  and  in  accuracy  and  con- 
venience does  not  compare  with  the  triangle  and  ruler  or  J  square. 


Fig.  40. 


SWEEPS  AND  VARIABLE  CURVES. 

For  drawing  circular  arcs  of  large  radius,  beyond  the  range  of  the  or- 
dinary compasses,  thin  slips  of  wood,  termed  sweeps,  are  usefully  employed, 

of  which  one  or  both  edges 
are  cut  to  the  required  circle. 
For  curves  which  are  not 
circular,  but  variously  ellip- 
tic or  otherwise,  "universal 
sweeps,"  made  of  thin  wood, 
of  variable  curvature,  are 
veiy  serviceable.  The  two 
examples  have  been  found 
from  experience  to  meet  almost  all  the  requirements  of  ordinary  drawing 


Fig.  42.    One  fourth  full  size. 


DRAWING  INSTRUMENTS.  11 

practice.     "Whatever  be  the  nature  of  the  curye,  some  portion  of  the  uni- 
versal sweep  will  be  found  to  coincide  with 
its  commencement,  and  it  can  be  continued 
throughout  its  extent  by  applying  successively 
such  parts  of  the  sweep  as  are  suitable,  taking 

care,  however,  that  the  continuity  is  not  in-          Fig  42    Or,^rth  full  siZ(T 
jured  by  unskilful  junction. 

ISTo  varnish  of  any  description  should  be  applied  to  any  of  the  wooden 
instruments  used  in  drawing,  as  the  best  varnish  will  retain  dust,  and  soil 
the  paper.  Use  the  wood  in  its  natural  state,  keeping  it  carefully  wiped. 
Various  other  materials  besides  wood  have  been  used,  as  steel  for  the  blades 
of  the  T  square  and  the  ruler ;  the  objection  is  the  liability  to  soil  the 
paper.  Glass  is  frequently  used  for  the  ruler  and  the  triangle,  and  retains 
its  correctness  of  edge  and  angle,  but  it  is  too  heavy,  and  liable  of  course 
to  fracture. 


THE    C.OMP  ASSES     OR    DIVIDERS. 

The  best  compasses  are  constructed  with  joints  of  two  different  metals, 
as  steel  and  brass,  whereby  the  wear  is  more  equal,  and  the  motion  of  the 
legs  uniform  and  steady,  and  not  subject  to  sudden  jerks  in  opening  or 
shutting.  This  motion  will  occasionally  require  some  adjustment  to  render 
it  uniformly  smooth,  and  to  move  stiffer  or  easier  at  pleasure,  but  so  that 
they  may  keep  steadily  any  position  that  may  be  given  to  them.  This  ad- 
justment is  performed  by  the  application  of  a  turnscrew  to  the  axis  of  the 
joint.  In  the  common  compasses,  a  simple  screw  forms  the  axis,  which 
may  be  turned  with  a  screwdriver ;  but  in  the'  best  made  instruments,  a 
steel  pin  passes  through  the  joints,  having  at  one  end  a  head  of  brass 
riveted  fast  upon  it,  and  on  the  other  end  a  similar  plate  or  nut  is  screwed, 
on  a  diameter  of  which  are  drilled  two  small  holes  for  the  application  of  a 
key  (fig.  43).  The  points  of  a  well  made  instrument  should  be 
of  steel  so  tempered,  as  neither  to  be  easily  bent  or  blunted ;  not 
too  fine  and  tapering,  and  yet  meeting  closely  when  the  com- 
passes are  shut. 

Instruction  for  using  dividers,  which  are  applied  only  to 
measure  and  transfer  distances  and  dimensions,  may  appear      Fig.  43. 
superfluous;  but  there  are  a  few  simple  directions  which   may  save  the 
young  draughtsman  much  perplexity  .and  loss  of  time.    It  is,  TO  course,  de- 
sirable to  work  the  compasses  in  such  a.manner  that,  when  the  dimension 


12  DRAWING   INSTRUMENTS. 

is  taken,  it  may  suffer  no  disturbance  in  its  transfer  from  the  scale  to  the 
drawing.  In  order  to  this,  the  instrument  is  to  be  held  by  the  head  or 
joint,  the  forefinger  resting  on  the  top  of  the  joint,  and  the  thumb  and 
second  finger  on  either  side.  When  held  in  this  way,  there  is  no  pressure 
except  on  the  head  and  centre,  and  the  dimension  between  the  points  can- 
not be  altered ;  but  if  the  instrument  be  clumsily  seized  by  a  thumb  on 
one  leg,  and  two  fingers  on  the  other,  the  pressure,  in  the  act  of  transfer- 
ence, must  inevitably  contract,  in  some  small  degree,  the  opening  of  the 
compasses  ;  and  if  the  dimension  has  to  be  set  off  several  times,  the  proba- 
bility is,  that  no  two  transfers  will  be  exactly  the  same.  And  whilst  it  is  all 
important  to  keep  the  dimension  exact,  it  is  also  desirable  to  manipulate  in 
such  a  way,  when  setting  off  the  same  dimension  a  number  of  times,  that 
the  point  of  position  be  never  lost.  Persons  unaccustomed  to  the  use  of 
compasses,  are  very  apt  to  turn  them  over  and  over  in  the  same  direction, 
when  laying  down  a  number  of  equal  measures,  and  this  necessitates  a  fre- 
quent change  of  the  finger  and  thumb,  which  direct  the  movement  of  the 
instrument ;  the  consequence  is,  either  that  the  fixed  leg  is  driven  deep 
into  the  drawing,  or  it  loses  position.  Now,  if  the  movement  be  alternately 
above  and  below  the  line  on  which  the  distances  are  being  set  off,  the  coin- 
passes  can  be  worked  with  great  freedom  and  delicacy,  and 
without  any  liability  to  shifting.  If  a  straight  line  is  drawn,  and 
semicircles  be  described  alternately  above  and  below  the  line,  it 
will  show  the  path  of  the  traversing  foot.  If  the  two  movements 
are  tried,  the  superiority  of  the  one  recommended  will  at  once  be 
discovered.  The  forefinger  rests  gently  on  the  head ;  and  the 
thumb  and  second  finger,  without  changing  from  side  to  side, 
direct  the  movement  for  setting  off  any  number  of  times  that 
may  be  required. 

The  hair  com/passes  (fig.  44)  are  constructed  in  the  same  man- 
ner as  the  common  compasses.  The  only  difference  consists  in  a 
contrivance,  whereby  the  lower  or  point  half  of  one  shank  can 
be  moved  a  very  small  quantity  either  .to wards  or  from  the  other 
point,  so  that  when  the  compasses  are  opened  nearly  to  the  re- 
quired extent,  by  the  help  of  the  screw  5  the  points  may  be  set 
with  great  precision,  which  cannot  be  done  so  well  by  the  motion 
of  the  joints  alone. 

Compasses  with  movable  points  (fig.  45)  are  a  pair  of  compasses 
^Jtorhich  the  point  half  of  one  of  the  legs  is  movable,  to  admit 
7is-  «•     of  adapting  singly  a  pen,  a  pencil,  or  a  dotting  point.     The  pen 
point  is  used  for  drawing  circles  or  arcs  with  ink.     The  pencil  point  is  a 


DRAWING  INSTRUMENTS. 


13 


Fig.  45. 

The  annexed  engraving 


a  tube  adapted  to  hold  a  piece  of  lead  pencil  for  describing  circles  or  arcs, 

and  the  dotting  point  consists  of  two  blades,  between  which  revolves  a 

small  wheel,  with  numerous  points  round  its 

circumference,  resembling  the  rowel  of  a  spur. 

The  space  between  the  blades  being  supplied 

with  Indian  ink,  as  the  compasses  describe  a 

circle  or  arc,  each  point,  as  the  wheel  revolves, 

will  pass  through  the  ink,  and  transfer  it  to 

the  paper  beneath,  making  equidistant  dots  in 

the  circle  which  the  compasses  describe. 

The  movable  points  have  a  joint  in  them, 
just  under  that  part  which  locks  into  the 
shank  of  the  compasses,  by  which  the  part  be- 
low the  joint  may  be  set  perpendicular  to  the 
plane  on  which  the  lines  are  described,  when 
the  compasses  are  open. 

An  additional  piece,  called  a  lengthening 
bar,  is  frequently  applied  to  these  compasses, 
to  enable  them  to  strike  larger  circles,  or  mea- 
sure greater  extents  than  they  otherwise  could, 
represents  this  instrument  and  its  appendages. 

A,  the  compasses,  with  a  movable  point  at  B ;  C  and  D,  the  joints  to 
set  each  point  perpendicular  to  the  paper  ;  E,  the  pencil  point ;  F,  the  pen 
point ;  G,  the  lengthening  bar. 

Bow  compasses. — These  are  a  small  pair,  either  having  a 
point  for  ink  or  pencil,  used  to  describe  small  arcs  or  circles, 
which  they  do  more  conveniently  than  large  compasses.     Fig. 
46  is  adapted  for  describing  arcs  of  a  radius  intermediate  between 
those  described  by  the  above-named  compasses,  and 
those  capable  of  being  produced  by  the  bows  repre- 
sented by  fig.  47.     In  fig.  46,  the  legs  can  be  opened  a 
considerable  width  by  the  joint,  whilst  in  fig.  47,  the 
opening  is  limited,  the  two  blades  or  legs  being  formed 
out  of  one  solid  piece  of  steel,  and  tempered  so  as  to 
form  a  spring  at  the  upper  part ;  the  spring  of  the  two 
blades  is  then  kept  in  obedience  by  an  adjusting  screw 
D,  by  which  the  two  points  may  be  set  to  any  required 
degree  of  minuteness,  and  very  small  circles  may  be 
described  with  precision.  F1s-  46-      Fis- 4T- 

The  pen  bows  (figs.  48,  49)  are  similar  in  their  construction  to  the  pen- 


DRAWING    INSTRUMENTS. 


Fig.  4a        Fig.  49. 


cil  bows.  In  fig.  49  there  is  a  second  joint  A,  by  which,  when  the  instru- 
•  ment  is  open  for  use,  the  pen  may  be  set  perpendicular,  or  nearly  so,  to 
the  paper,  which  is  essential  in  the  use  of  the  draw- 
ing pen. 

Similar  to  fig.  48  in  their  construction  are  the 
spring  dividers  (fig.  50),  particularly  useful  for  re- 
peating divisions  of  a  small  but  equal  extent,  a 
practice  that  has  acquired  the  name  of  stepping. 

The  drawing  pen  (fig.  51)  is  used  for  drawing 
straight  lines.  It  consists  of  two  blades  with  steel 
points  fixed  to  a  handle ;  and  they  are  so  bent,  that  a 
sufficient  cavity  is  left  between  them  for  the  ink,  when 
the  ends  of  the  steel  points  meet  close  together,  or 
nearly  so.  The  blades  are  set  with  the  points  more 
or  less  open  by  means  of  a  millheaded  screw,  so  as 
to  draw  lines  of  any  required  fineness  or  thickness. 
Fig.  50.  One  of  the  blades  is  framed  with  a  joint,  so  that  by 
taking  out  the  screw,  the  blades  may  be  completely  opened,  and  the  points 
effectively  cleaned  after  use.  The  ink  is  to  be  put  between  the  blades  by 
a  common  pen,  and  in  using  the  pen  it  should  be 
slightly  inclined  in  the  direction  of  the  line  to  be 
drawn,  and  care  should  be  taken  that  both  points 
touch  the  paper ;  and  these  observations  equally  apply 
to  the  pen  points  of  the  compasses  before  described. 
The  drawing  pen  should  be  kept  close  to  the  ruler 
or  straight  edge,  and  in  the  same  direction  during 
the  whole  operation  of  drawing  the  line.  Care 
must  be  taken  in  holding  the  straight  edge  firmly 
with  the  left  hand,  that  it  does  not  change  its  posi- 
tion. 

For  drawing  close  parallel  lines  in  mechanical 
and  architectural  drawings,  or  to  represent  canals 
or  roads,  a  double  pen  (fig.  52)  is  frequently  used, 
with  an  adjusting  screw  to  set  the  pen  to  any  re- 
quired small  distance.  This  is  usually  called  the 
road  pen.  The  best  pricking  point  is  a  fine  needle 
held  in  a  pair  of  forceps  (fig.  53).  It  is  used  to  mark 
the  intersection  of  lines,  or  to  set  off  divisions  from 
the  plotting  scale  and  protractor.  This  point  may 
Fig.  51.  Fig.  52.  rig.  53.  also  be  used  to  Prick  through  a  drawing  upon  an 


X 


DRAWING-   ESTSTRUMEIsrTS.  15 

intended  copy,  or,  the  needle  being  reversed,  the  eye  end  forms  a  good 
tracing  point. 

For  filling  up  the  broad  lines  of  borders,  a  goose  quill  is  often  used 
with  a  short  nib  and  no  slit  (fig.  5-i).     In  drawing  with  this  pen,  incline 


Fig.  54. 

the  drawing-board  so  that  the  ink  will  follow  the  pen,  which  prevents  blots 
or  the  accumulation  of  too  much  ink  at  any  one  point. 

The  dotting  point  (fig.  55)  resembles  a  drawing  pen,  except  that  the 
points  are  not  so  sharp.     On  the  back  blade,  as  seen  in  the  engraving,  is  a 
pivot,  on  which  may  be  placed  a  dotting  wheel,  a,  resembling  the 
rowel  of  a  spur ;  the  screw  5  is  for  opening  the  blades  to  remove  the    \— jc 
wheel  for  cleaning  after  use,  or  replacing  it  with  one  of  another 
character  of  dot.     The  cap  c,  at  the  upper  end  of  the  instrument,  is 
a  box  containing  a  variety  of  dotting  wheels,  each  producing  a  dif- 
ferent shaped  dot.     These  are  used  as  distinguishing  marks  for  dif- 
ferent classes  of  boundaries  on  maps  ;  for  instance,  one  kind  of  dot 
distinguishes  county  boundaries,  another  kind  town  boundaries,  a 
third  kind  distinguishes  that  which  is  both  a  county  and  a  town 
boundary,  &c.,  &c.     In  using  this  instrument,  the  ink  must  be  in- 
serted between  the  blades  above  the  dotting  wheel,  so  that,  as  the 
wheel  revolves,  the  points  shall  pass  through  the  ink,  each  carrying 
with  it  a  drop,  and  marking  the  paper  as  it  passes.     It  sometimes 
happens  that  the  wheel  will  revolve  many  times  before  it  begins  to 
deposit  its  ink  on  the  drawing,  thereby  leaving  the  first  part  of  the 
line  altogether  blank,  and  in  attempting  to  go  over  it  again,  the  first 
made  dots  are  liable  to  get  blotted.     This  evil  may  be  mostly  reme- 
died by  placing  a  piece  of  blank  paper  over  the  drawing  to  the  very  point 
the  dotted  line  is  to  commence  at,  then  begin  with  drawing  the  wheel  over 
the  blank  paper  first,  so  that  by  the  time  it  will  have  arrived  at  the  proper 
point  of  commencement,  the  ink  may  be  expected  to  flow  over  the  points 
of  the  wheel,  and  make  the  dotted  line  perfect  as  required. 

Drawing  pins  (fig.  56)  are  used  to  hold  paper  down  upon  a 
drawing  or  other  board  in  any  required  position,  and  in  most 
cases  answer  better  than  heavy  weights,  which  are  frequently 
used  for  that  purpose,  as  the  board  may  be  shifted  from  place  Fig'56t 


DRAWING   INSTRUMENTS. 


to  place  without  moving  the  paper.  They  consist  of  a  brass  head,  with 
a  steel  point  at  right  angles  to  its  plane.  A  represents  it  as  seen  edgewise, 
and  B  as  seen  from  above. 


SCALES. 

Fig.  57  represents  the  usual  scale  to  be  found  in  the  common  boxes  of 
drawing  instruments.  It  contains,  on  its 
two  sides,  simply  divided  scales,  a  diagonal 
scale  and  a  protractor.  The  simply  divided 
scales  consist  of  a  series  of  equal  divisions  of 
an  inch,  which  are  numbered  1,  2,  3,  &c., 
beginning  from  the  second  division  on  the 
left  hand. 

It  will  be  seen  (figure  58)  that  the  dif- 
ferent scales  are  marked  30,  35,  40,  &c.,  and 
that  the  upper  part  of  the  left  division  in 
eat;h  is  subdivided  into  twelve  equal  parts, 
and  that  the  lower  part  of  the  same  division 
is  subdivided  into  ten  equal  parts.  If  now 
these  last  subdivisions  or  tenths  be  consid- 
ered as  units,  one  mile,  or  one  chain,  or  one 
foot,  then  each  primary  division  will  repre- 
sent ten  units,  ten  miles,  ten  chains,  or  ten 
feet,  and  the  scale  is  said  to  be  30,  35,  40  (ac- 
cording to  the  scale  selected)  miles,  chains, 
or  feet  to  the  inch.  Tims,  suppose  that  it 
were  required  on  a  scale  of  30  feet  to  the 
inch,  to  lay  off  47  feet.  On  the  scale  marked 
30,  place  one  point  of  the  compasses  or  di- 
viders at  4,  and  bring  the  other  point  to  the 
7th  lower  subdivisions,  counting  from  the 
right,  and  we  have  the  distance  required. 
Each  of  the  primary  divisions  may  be  re- 
garded as  unit,  one  foot  for  instance ;  then 
the  upper  subdivisions  are  twelfths  of  a  foot 
or  inches,  and  the  lower  subdivisions  tenths 
of  an  inch, 
rig.  ST.  In  fig.  57,  the  scales  are  marked  at  the 


DRAWING  INSTUrMK.NTS. 


17 


left,  1  in.  i,  •£,  i,  but  the  divisions  and  subdivisions  are  as  above.  In  this 
fig.,  the  primary  divisions  are  one  inch,  f ,  •£,  and  J  of  an  inch.  These  scales 
are  more  generally  used  for  drawings  of  machinery  and  of  architecture, 


ki  i,  •vriojqrrrEi 


i      IT 


i   I  .1   II   I 


LA. 


17 


17 


Fig.  58. 

while  fig.  58  are  for  topographical  drawings.  The  application  of  these 
scales  are  similar  to  those  already  described.  When  the  primary  divisions 
are  considered  inches,  then  the  drawings  will  be  each  full,  f ,  £,  or  J  size, 
according  to  the  scale  adopted. 

On  the  selection  of  the  scale. — In  all  working  architectural  and  me- 
chanical drawings,  use  as  large  a  scale  as  possible  ;  neither  depend,  even 
in  that  case,  that  the  mechanics  employed  in  the  construction  will  measure 
correctly,  but  write  in  the  dimensions  as  far  as  practicable.  For  architec- 
tural plans,  the  scale  of  \  an  inch  to  the  foot  is  one  of  very  general  use, 
and  convenient  for  the  mechanic,  as  the  common  two-foot  rule  carried  by 
all  mechanics  is  subdivided  into  iths,  |ths,  and  sometimes  sixteenths  of  an 
inch,  and  the  distances  on  a  drawing  to  this  scale  can  therefore  be  easily 
measured  by  them.  This  fact  should  not  be  lost  sight  of  in  working  draw- 
ings. When  the  dimensions  are  not  written,  make  use  of  such  scales  that 
the  distances  may  be  measured  by  the  division  of  the  common  two-foot 
rule  ;  thus,  in  a  scale  of  \  or  \  full  size,  6  inches  or  3  inches  represent 
one  foot ;  in  a  scale  of  an  inch  to  the  foot  or  twelfth  full  size,  each  \  an 
inch  represents  6  inches,  \  3  inches  ;  but  when  \  or  -j-1^  an  inch  to  the  foot, 
or  any  similar  scale,  is  adopted,  it  is  evident  that  these  divisions  cannot 
be  taken  by  the  two-foot  rule.  The  scale  should  be  written  on  every 
drawing,  or  the  scale  itself  should  be  drawn  on  the  margin.  In  topographi- 
cal and  geodesic  drawings  the  latter  is  essential,  as  the  scale  adopted  fre- 
quently has  to  be  drawn  for  the  specific  purpose,  and  the  paper  itself  con- 
tracts or  expands  with  every  atmospheric  change,  and  the  measurements 
will  therefore  not  agree  at  all  times  with  a  detached  scale  ;  and  moreover, 
a  drawing  laid  down  from  such  detached  scale,  of  wood  or  ivory,  will  not 
be  uniform  throughout,  for  on  a  damp  day  the  measurements  will  be  too 
short,  and  on  a  dry  day  too  long.  Mr.  Holtzapffel  has  sought  to  remedy 
this  inconvenience  by  the  introduction  of  paper  scales  ;  but  all  kinds  of  paper 
do  not  contract  and  expand  equally,  and  the ,  error  is  therefore  only  par- 
tially corrected  by  his  ingenious  substitution  of  one  material  for  another. 


18 


DRAWING   INSTRUMENTS. 


5±FFP* 

»  B 

fffl  

m— 

Hi  

m£~~~ 

±3  

rj_ 

dT 

rW- 

J) 


1 

Fig.  59. 


Diagonal  scales. — The  simply  divided  scales  give  only  two  denomina- 
tions, primaries  and  tenths,  or  twelfths ;  but  more  minute  subdivision  is 
attained  by  the  diagonal  scale,  which  consists  of  a  number  of  primary 
divisions,  one  of  which  is  divided  into  tenths,  and  subdivided  into  hun- 

dredths  by  diagonal 
lines  (fig.  59).  This 
scale  is  constructed  in 
the  following  manner : 
— Eleven  parallel  lines 
are  ruled,  enclosing  ten 
equal  spaces  ;  the  length  is  set  off  into  equal  primary  divisions,  as  DE.  El, 
&c. ;  the  first  DE  is  subdivided,  and  diagonals  are  then  drawn  from  the  sub- 
divisions between  A  and  B,  to  those  between  D  and  E,  as  shown  in  the  dia- 
gram. Hence  it  is  evident  that  at  every  parallel  we  get  an  additional  tenth 
of  the  subdivisions,  or  a  hundredth  of  the  primaries,  and  can  therefore  obtain 
a  measurement  with  great  exactness  to  three  places  of  figures.  To  take  a 
measurement  of  (say)  168,  we  place  one  foot  of  the  dividers  on  the  primary 
1,  and  carry  it  down  to  the  ninth  parallel,  and  then  extend  the  other  foot 
to  the  intersection  of  the  diagonal,  which  falls  from  the  subdivision  6,  with 
the  parallel  that  measures  the  eight-hundredth  part  (fig.  60).  The  pri- 
maries may  of  course  be  considered  as  yards,  feet,  or  inches  ;  and  the  sub- 

n     „    \     3      , 2       divisions  as  tenths  and 

J  hundredths  of  these 
respective  denomina- 
tions. 

The  diagonals  may 
be  applied  to  a  scale 
where  only  one  sub- 
division is  required. 
Thus,  if  seven  lines  be 
(fig.  61)  ruled,  enclos- 
ing six  equal  spaces,  and  the  length  be  divided  into  primaries,  as  AB,  BC, 
&c.,  the  first  primary  AB  may  be  subdivided  into  twelfths  by  two  diagonals 
A.  G  s  (T  running  from  6,  the  mid- 

dle of  AB,  to  12  and  0. 
We  have  here  a  very  con- 
venient scale  of  feet  and 
inches.  From  C  to  6  is 
1  foot  6  inches ;  and  from 


Fig.  60. 


7/V 

-4-  v 

•••»/           \x 

JO/                   \2 

»/.                        \, 

'          US 

0/2 

DRAWING    INSTRUMENTS. 


1!) 


C  on  the  several  parallels  to  the  various  intersections  of  the  diagonals,  we 
obtain  1  foot  and  any  number  of  inches  from  1  to  12. 

Plotting  scales  and  rulers  are  scales  of  equal  parts,  with  the  divisions 
placed  on  a  fiducial  edge,  by  which' any  length  may  be  pricked  off  on  to  the 
paper  without  using  the  compasses,  whose  points,  by  frequent  use,  destroy 
the  fineness  of  the  graduation. 

On  the  scale  (fig.  57)  in  common  boxes  of  drawing  instruments,  the 
edge  of  one  side  is  divided  as  a  protractor,  for  the  laying  out  of  angles. 
The  instrument,  when  by  itself,  consists  of  a  semicircle  of  thin  metal  or 


horn  (fig.  62),  whose  circumference  is  divided  into  180  equal  parts  or  de- 
grees (180°).  In  the  larger  protractors  each  of  these  divisions  is  sub- 
divided. 

Application  of  the  protractor. — To  lay  off  a  given  angle  from  a  given 
point  on  a  straight  line,  let  the  straight  line  a  J)  of  the  protractor  coincide 
with  the  given  line,  and  the  point  c  with  the  given  point ;  now  mark  on 
the  paper  against  the  division  on  the  periphery,  coinciding  with  the  angle 
required  ;  remove  the  protractor,  and  draw  a  line  through  the  given  point 
and  the  mark. 

The  instruments  already  described  are  those  to  be  found  in  the  usual 
cases  of  drawing  instruments,  and  are  sufficient  for  all  the  ordinary  pur- 
pose of  draughtsmen  ;  but  there  are  others  adapted  to  special  purpose,  or  of 
careful  and  elaborate  workmanship,  which  are  useful  where  great  accuracy 
and  finish  are  required,  and  of  some  of  which  descriptions  will  be  given. 

Vernier  scales  are  preferred  by  some  to  the  diagonal  scale  already 
described.  To  construct  a  vernier  scale  by  which  a  number  to  three 
places  may  be  taken,  divide  all  the  primary  divisions  into  tenths,  and 


20  DRAWING  INSTRUMENTS. 

number  these  subdivisions  1,  2,  3,  from  left  to  right.  Take  off  now 
with  the  compasses  eleven  of  these  subdivisions,  set  the  extent  off  back- 
wards from  the  end  of  the  first  primary  division,  and  it  will  reach  beyond 
the  beginning  of  this  division,  or  zero  point,  a  distance  equal  to  one  of  the 
subdivisions.  Now  divide  the  extent  thus  set  off  into  ten  equal  parts, 
marking  the  divisions  on  the  opposite  side  of  the  divided  line  to  the  strokes 
marking  the  primary  divisions  and  the  subdivisions,  and  number  them  1, 
2,  3,  &c.,  backwards  from  right  to  left.  Then,  since  the  extent  of  eleven 
subdivisions  has  been  divided  into  ten  equal  parts,  so  that  these  ten  parts 
exceed  by  one  subdivision  the  extent  of  ten  subdivisions,  each  one  of  these 
equal  parts,  or,  as  it  may  be  called,  one  division  of  the  vernier  scale,  ex- 
ceeds one  of  the  subdivisions  by  a  tenth  part  of  a  subdivision,  or  a  hun- 
dredth part  of  a  primary  division ;  thus,  if  the  subdivision  be  considered 
10,  then  frorii  0  to  the  first  division  of  the  vernier  will  be  11 ;  to  the  sec- 
ond, 22 ;  to  the  third,  33 ;  to  the  fourth,  M;  to  the  fifth,  55 ;  and  so  on, 
66,  77,  88,  99. 


10 

»  1  1  1 

8       1 

—  rn  —  i  —  i  —  1  —  i  —  i  —  i  —  i  — 

—  i  —  i  —  i  —  i  i  i  —  TT  —  i  — 

—  i  —  i  —  i  —  i  —  1  —  i  —  i  —  i  —  i  — 

Ann  I— 

1     l     1     1    ( 

*Pg_J         Si    *         2 

Fig.  63. 

To  take  off  the  number  253  from  this  scale,  place  one  point  of  the  di- 
viders at  the  third  division  of  the  vernier ;  if  the  other  point  be  brought  to 
the  primary  division  2,  the  distance  embraced  by  the  dividers  will  be  233, 
and  the  dividers  must  be  extended  to  the  second  subdivision  of  tenths  to 
the  right  of  2.  If  the  number  were  213,  then  the  dividers  would  have  to 
be  closed  to  the  second  subdivision  of  tenths  to  the  left  of  2.  To  take  off 
the  number  59  from  the  scale,  place  one  point  of  the  dividers  at  the  ninth 
division  of  the  vernier ;  if  the  other  point  be  extended  to  the  0  mark,  the 
dividers  will  embrace  99,  and  must  therefore  be  closed  to  the  fourth  subdi- 
vision to  the  left  of  0. 

These  numbers,  thus  taken,  may  be  253,  25-3,  2*53 ;  213,  21*3,  2*13  ;  59, 
5*9,  .59,  according  as  the  primary  divisions  are  taken  as  hundreds,  tens,  or 
units. 

The  construction  of  this  scale  is  similar  to  that  of  the  verniers  of  theod- 
olites and  surveying  instruments ;  but,  in  its  application  to  drawing,  is  not 
as  simple  as  the  diagonal  scales,  figs.  59,  61. 

On  some  of  the  plain  scales  in  the  instrument  boxes  will  be  found  divi- 
sions marked  as  in  fig.  64.  Many  of  the  divisions  here  laid  down  have  no 
application  to  drawing,  according  to  the  scope  of  this  work ;  a  brief  ex- 
planation and  application  will  therefore  only  be  given.  Under  definitions 


DRAWING   INSTRUMENTS. 


and  technicalities,  the  signification  of  the  terms  chords,  tangents,  sines,  and 
secants,  has  been  defined.     The  chord  of  60°  is  equal  to  radius,  or  half 


Rim 

Laii 
CIw 
Siit, 
Tan 


S\) 


Fig.  65. 


Fig.  64 

the  diameter.     The  line  of  chords  is  used  to  set  off  an  angle,  or  to  measure 
an  angle  already  laid  down. 

To  set  off  an  angle. — An  angle  of  35°  for  instance  :  open  the  compasses 
to  the  extent  of  60°  on  the  scale  of  chords,  setting  one  point  at  A  on  the 
line  A  B,  describe  with  the  other  point  an  arc  ;  again  with  the  compasses 
open  to  the  extent  of  35°  on  the  scale,  setting  one  point  on  B,  describe  an 
arc,  cutting  the  arc  B  C ;  through  this  intersection  and  the  point  A, 
draw  the  line  A  C,  and  we  have  the  angle  CAB,  35°. 

To  measure  the  angle  contained  by  the  straight  lines  A  B  and  A  C  al- 
ready laid  down.  ,  Open  the  compasses  to  the  extent 
of  60°  on  the  line  of  chords,  as  before,  and  with  this 
radius  describe  the  arc  B  C,  cutting  A  B  and  A  C, 
produced,  if  necessary,  in  the  points  B  and  C  ;  then, 
extending  the  compasses  from  B  to  0,  place  one 
point  of  the  compasses  on  the  beginning  or  zero 
point,  of  the  line  of  chords,  and  the  other  point  will  extend  to  the  number 
upon  this  line,  indicating  the  degrees  in  the  angle  BAG. 

The  lines  of  sines,  secants,  tangents,  and  semitangents  are  principally 
used  for  the  several  projections,  or  perspective  representations,  of  the  circles 
of  the  sphere,  by  means  of  which  maps  are  constructed. 

The  line  of  rhumbs  is  a  scale  of  the  chords  of  the  angles  of  deviation 
from  the  meridian  denoted  by  the  several  points  and  quarter  points  of  the 
compass,  enabling  the  navigator,  without  computation,  to  lay  down  or 
measure  a  ship's  course  upon  a  chart. 

The  line  of  longitudes  shows  the  number  of  equatorial  miles  in  a  de- 
gree of  longitude  on  the  parallels  of  latitude  indicated  by  the  degrees  on 
the  corresponding  points  of  the  line  of  chords.  Example. — A  ship  in  lati- 
tude 60°  K.  sailing  E.  Y9  miles,  required  the  difference  of  longitude  be- 
tween the  beginning  and  end  of  her  course.  Opposite  60  on  the  line  of 
chords  stands  30  on  the  line  of  longitudes,  which  is,  therefore,  the  number 
of  equatorial  miles  m  a  degree  of  longitude  at  that  latitude.  Hence,  ||= 
2°  38',  the  required  difference  of  longitude. 


DRAWING   INSTRUMENTS. 


The  sector  (fig.  66)  consists  of  two  flat  rulers  united  by  a  central  joint, 
and  opening  like  a  pair  of  compasses.     It  carries  several  plain  scales  on  its 
faces,  but  its  most  important  lines  are  in  pairs, 
running  accurately  to  the  central  joint. 

Plain  scales  on  the  sector. — On  the  outer  edge 
of  the  sector  is  usually  given  a  decimal  scale  from 
1  to  100  ;  and  in  connection  with  it,  on  one  of  the 
sides,  a  scale  of  inches  and  tenths.  These  are 
identical  with  the  lines  on  the  plain  scale,  previ- 
ously mentioned,  but  the  latter  are  more  commo- 
diously  placed  for  use.  On  the  other  side  we  have 
logarithmic  lines  of  numbers,  sines,  and  tangents. 

Sectoral  double  scales. — These  are  respectively 
named  the  Lines  of  Lines,  Chords,  Secants,  Sines, 
and  Tangents.  These  scales  have  one  line  on  each 
ruler,  and  the  two  lines  converge  accurately  in  the 
central  joint  of  the  sector. 

The  principle  on  which  the  double  scales  are 
constructed  is,  that  similar  triangles  have  their 
like  sides  proportional.  Let  the  lines  A  B,  A  C, 
represent  the  legs  of  the  sector,  and 
A  D,  A  E,  two  equal  sections  from 
the  centre  ;  then,  if  the  points  B  C 
and  D  E  be  connected,  the  lines  B  C 
and  D  E  will  be  parallel ;  therefore, 
the  triangles  A  B  C,  A  D  E,  wiU  be 
similar,  and  consequently  the  sides 
A  B,  B  C,  A  D,  D  E,  proportional, 
that  is,  as  A  B  :  B  C  : :  A  D  :  D  E  ; 
so  that  if  A  D  be  the  half,  third,  or 
fourth  part  of  A  B,  then  D  E  will 
be  a  half,  third,  or  fourth  part  of 
B  C  ;  and  the  same  holds  of  all  the 
rest.  Hence,  if  D  E  be  the  chord,  sine,  or  tangent 
Fig.  ee.  of  any  arc,  or  of  any  number  of  degrees  to  the 

radius  A  D,  then  B  C  will  be  the  same  to  the  radius  A  B.  Thus  at  every 
opening  of  the  sector,  the  transverse  distances  D  E  and  C  B  from  one  ruler 
to  another,  are  proportional  to  the  lateral  distances,  measured  on  the  lines 
A  B,  A  C.  It  is  to  be  observed,  that  all  measures  are  to  be  taken  from 
the  inner  lines,  since  these  only  run  accurately  to  the  centre. 


DRAWING   INSTRUMENTS.  23 

The  line  of  lines,  marked  L  on  each  leg  of  the  sector. — This  is  a  line 
of  10  primaries,  each  subdivided  into  tenths,  thus  making  100  divisions. 
Its  use  is,  to  divide  a  given  line  into  any  number  of  equal  parts ;  to  give 
accurate  scale  measures  for  the  construction  of  a  drawing ;  to  form  any 
required  scale ;  to  divide  a  given  line  in  any  assigned  proportion ;  and  to 
find  third,  fourth,  and  middle  proportionals  to  given  right  lines. 

To  divide  a  given  line  into  eight  equal  parts. — Take  the  line  in  the  com- 
passes, and  open  the  sector  so  as  to  apply  it  transversely  to  8  and  8,  then 
the  transverse  from  1  to  1  will  be  the  eighth  part  of  the  line. 

To  form  any  required  scale  of  equal  parts. — Take  one  inch  in  the  com- 
passes, and  open  the  sector,  till  this  extent  becomes  a  transverse  distance 
at  the  division  indicating  the  number  of  parts  in  an  inch  of  the  required 
scale. 

Example. — To  adjust  the  sector  as  a  scale  of  one  inch  to  four  chains. 
— Make  one  inch  the  transverse  distance  of  4  and  4 ;  then  the  transverse 
distances  of  the  other  corresponding  divisions  and  subdivisions  will  repre- 
sent the  number  of  chains  and  links  indicated  by  these  divisions  :  thus,  the 
transverse  distance  from  3  to  3  will  represent  three  chains. 

To  construct  a  scale  of  feet  and  inches  in  such  a  manner,  that  an  ex- 
tent of  three  inches  shall  represent  twenty  inches. — Make  three  inches  a 
transverse  distance  between  10  and  10,  and  the  transverse  distance  of  6  and 
6  will  represent  12  inches.  Set  off  this  extent,  divide  it  into  12  equal  parts, 
each  of  these  divisions  will  represent  an  inch.  Place  the  figure  0  at  the 
right,  and  set  off  again  the  extent  of  the  whole  twelve  parts,  from  0  to  1, 
1  to  2,  &c.,  to  represent  the  feet. 

Proportion. — Two  lines  being  given,  to  find  a  third  proportional. 

Example. — The  given  lines  =  2  and  6,  a  third  proportional  required. 
Take  between  the  compasses  the  lateral  distance  of  the  second  term  6  on  any 
convenient  scale,  and  open  the  sector  until  this  distance  becomes  the  trans- 
verse distance  to  the  first  term  2  ;  then  the  transverse  distance  of  the  second 
term  6,  measured  upon  the  same  scale  as  the  former,  will  equal  18,  the 
third  proportional  required. 

Example. — to  find  a  fourth  proportional  to  the  numbers  2,  6,  and  10. 

Take  the  lateral  distance  of  the  second  term  6,  from  any  convenient 
scale  of  equal  parts,  and  open  the  sector  until  that  quantity,  or  any  aliquot 
part  thereof,  becomes  the  transverse  distance  of  the  first  term  2,  then  the 
transverse  distance  of  the  third  term  10,  taken  from  the  same  scale  of  equal 
parts,  will  give  30,  the  fourth  proportional  required. 

Line  of  Chords,  marked  C  on  each  leg  of  the  sector.  The  double  scales 
of  chords  upon  the  sector  are  more  useful  than  the  single  line  of  chords  de- 


24  DRAWING   INSTRUMENTS. 

scribed  on  the  plane  scale ;  for  on  the  sector,  the  radius  with  which  the  arc 
is  to  be  described  may  be  of  any  length  less  than  the  transverse  distance 
of  60  and  60  when  the  legs  are  opened  as  far  as  the  instrument  will  admit 
of.  But  with  the  chords  on  the  plane  scale,  the  arc  described  must  be 
always  of  the  same  radius. 

To  protract  an  angle  BAG,  which  shall  contain  a  given  number  of  de- 
grees, suppose  36°. 

Make  the  transverse  distance  of  60  and  60  equal  to  the  length  of  the 
radius  of  the  circle,  and  with  that  opening  de- 
scribe the  arc  B  C. 

Take  the  transverse  distance  of  the  given  de- 
grees 36,  and  lay  this  distance  on  the  arc  from 
the  point  B  to  C. 

From  the  centre  A  of  the  arc,  draw  A  C, 
rig.  68.  A  B,  and  these  two  lines  will  contain  the  angle 

required. 

To  protract  an  angle  of  more  than  60°,  divide  the  required  angle  by  2 
or  3,  and  set  off  as  above  twice  or  thrice  the  arc. 

From  what  has  been  said  about  the  protracting  of  an  angle  to  contain 
a  given  number  of  degrees,  it  will  easily  be  seen  how  to  find  the  degrees 
(or  measure)  of  an  angle  already  laid  down. 

Line  of  Polygons. — The  line  of  polygons  is  chiefly  useful  for  the  ready 
division  of  the  circumference  of  a  circle  into  any  number  of  equal  parts 
from  4  to  12 ;  that  is,  as  a  ready  means  to  inscribe  regular  polygons  of  any 
given  number  of  sides,  from  4  to  12,  within  a  given  circle.  To  do  which, 
set  off  the  radius  of  the  given  circle  (which  is  always  equal  to  the  side  of 
an  inscribed  hexagon)  as  the  transverse  distance  of  6  and  6  upon  the  line 
of  polygons.  Then  the  transverse  distance  of  4  and  4  will  be  the  side  of  a 
square ;  the  transverse  of  5  and  5  the  side  of  a  pentagon. 

If  it  be  required  to  form  a  polygon,  upon  a  given  right  line  set  off  the 
extent  of  the  given  line,  as  a  transverse  distance  between  the  points  upon 
the  line  of  polygons,  answering  to  the  number  of  sides  of  which  the  poly- 
gon is  to  consist,  as  for  a  pentagon  between  5  and  5,  or  for  an  octagon  be- 
tween 8  and  8 ;  then  the  transverse  distance  between  6  and  6  will  be  the 
radius  of  a  circle,  whose  circumference  would  be  divided  by  the  given  line 
into  the  number  of  sides  required. 

All  regular  polygons,  whose  number  of  sides  will  exactly  divide  360 
(the  number  of  degrees  into  which  all  circles  are  supposed  to  be  divided) 
without  a  remainder,  may  likewise  be  set  off  upon  the  circumference  of  a 
circle  by  the  line  of  chords.  Thus,  take  the  radius  of  the  circle  between 


DRAWING   INSTRUMENTS.  25 

the  compasses,  and  open  the  sector  till  that  extent  becomes  the  transverse 
distance  between  60  and  60  upon  the  line  of  chords ;  then  having  divided 
360  by  the  required  number  of  sides,  the  transverse  distance  between  the 
numbers  of  the  quotient  will  be  the  side  of  the  polygon  required.  Thus, 
for  an  octagon,  take  the  distance  between  45  and  45 ;  and  for  a  polygon 
of  36  sides,  take  the  distance  between  10  and  10,  &c. 

Lines  of  sines,  tangents  and  secants. — Given,  the  radius  of  a  circle, 
required  the  sine  and  tangent  of  28°  30'  to  that  radius — 

Open  the  sector,  so  that  the  transverse  distance  of  90  and  90  on  the 
sines,  or  of  45  and  45  on  the  tangents,  may  be  equal  to  the  given  radius ; 
then  will  the  transverse  distance  of  28°  30',  taken  from  the  sines,  be  the 
length  of  that  sine  to  the  given  radius  ;  or  if  taken  from  the  tangents,  will 
be  the  length  of  that  tangent  to  the  given  radius. 

But  if  the  secant  of  28°  30'  was  required — 

Make  the  given  radius  a  transverse  distance  of  0  and  0,  at  the  begin- 
ning of  the  line  of  secants,  and  then  take  the  transverse  distance  of  the  de- 
grees wanted,  viz.,  28°  30'. 

A  tangent  greater  than  45  degrees  (suppose  60)  is  found  thus : 

Make  the  given  radius  a  transverse  distance  to  45,  and  45  at  the  begin- 
ning of  the  scale  of  upper  tangents,  and  then  the  required  degrees  (60)  may 
be  taken  from  the  scale. 

Given  the  length  of  the  sine,  tangent,  or  secant  of  any  degrees,  to  find 
the  length  of  the  radius  to  that  sine,  tangent,  or  secant. 

Make  the  given  length  a  transverse  distance  to  its  given  degrees  on  its 
respective  scale.  Then 

If  a  sine,                           )  (  90  and  90  on  the  sines  )  will  be 

If  a  tangent  under  45°,  (  the  transverse  )  45  and  45  on  the  tangents  (  the  ra- 

If  a  tangent  above  45°,  f  distance  of  J  45  and  45  on  the  upper  tangents  f  dius 

If  a  secant,  (  0  and  0  on  the  secants  )  sought. 

To  find  the  length  of  a  versed  sine  to  a  given  number  of  degrees,  and  a, 
given  radius. 

Make  the  transverse  distance '  of  90  and  90  on  the  sine  equal  to  the 
given  radius.  Take  the  transverse  distance  of  the  complement  of  the  sine 
of  the  given  number  of  degrees.  If  the  given  number  of-  degrees  is  less 
than  90,  subtract  the  complement  of  the  sine  from  the  radius,  the  remain- 
der will  be  the  versed  sine. 

If  the  given  number  of  degrees  are  more  than  90,  add  the  complement 
of  the  sine  to  the  radius,  and  the  sum  will  be  the  versed  sine. 

To  open  the  legs  of  a  sector,  so  that  the  corresponding  double  scales  of 
lines,  chords,  sines,  tangents,  may  make  each  a  right  angle. 


£46  DRAWING   INSTRUMENTS. 

On  the  line  of  lines,  make  the  lateral  distance  10,  a  transverse  distance 
between  8  on  one  leg  and  6  on  the  other  leg. 

On  the  line  of  sines,  make  the  lateral  distance  90,  a  transverse  distance 
from  45  to  45,  or  from  40  to  50,  or  from  30  to  60,  or  from  the  sine  of  any 
degrees  to  their  complement. 

On  the  line  of  tangents,  make  the  lateral  distance  of  45  a  transverse 
distance  between  30  and  30. 

Marquois*s  scales  (fig.  69). — These  scales  consist  of  a  right-angled  tri- 
angle, of  which  the  hypothenuse  or  longest  side  is  three  times  the  length 
of  the  shortest,  and  a  rectangular  rule.  Our  figure,  which  is  drawn  one- 
third  the  actual  size  of  the  instruments  from  which  it  is  taken,  repre- 
sents the  triangle  and  a  rule,  as  being  used  to  draw  a  series  of  paral- 
lel lines.  The  rule  is  one  foot  long,  and  has,  parallel  to  each  of  its 
edges,  two  scales,  one  placed  close  to  the  edge,  and  the  other  immediately 


Fig.  69. 


within  this,  the  outer  being  termed  the  artificial,  and  the  inner  the  natural 
scale.  The  divisions  upon  the  outer  scale  are  three  times  the  length  of 
those  upon  the  inner  scale,  so  as  to  bear  the  same  proportion  to  each  other 
that  the  longest  side  of  the  triangle  bears  to  the  shortest.  In  the  artificial 
scales,  the  zero  point  is  placed  in  the  middle  of  the  edge  of  the  rule,  and 
the  primary  divisions  are  numbered  both  ways  from  this  point  to  the  two 
ends  of  the  rule,  and  are  every  one  subdivided  into  ten  equal  parts,  each 
of  which  is,  consequently,  three  times  the  length  of  a  subdivision  of  the 
corresponding  natural  scale. 

The  triangle  has  a  short  line  drawn  perpendicular  to  the  hypothenuse 
near  the  middle  of  it,  to  serve  as  an  index  or  pointer  ;  and  the  longest  of 
the  other  two  sides  has  a  sloped  edge. 

To  draw  a  line  parallel  to  a  given  line,  at  a  given  distance  from  it. — 1. 
Having  applied  the  given  distance  to  the  one  of  the  natural  scales  which 
is  found  to  measure  it  most  conveniently,  place  the  triangle  with  its  sloped 


DRAWING   INSTRUMENTS.  27 

edge  coincident  with  the  given  line,  or  rather  at  such  small  distance  from 
it,  that  the  pen  or  pencil  passes  directly  over  it  when  drawn  along  this 
edge.  2.  Set  the  rule  closely  against  the  hypothenuse,  making  the  zero 
point  of  the  corresponding  artificial  scale  coincide  with  the  index  upon  the 
triangle.  3.  Move  the  triangle  along  the  rule,  to  the  left  or  right  accord- 
ing as  the  required  line  is  to  be  above  or  below  the  given  line,  until  the 
index  coincides  with  the  division  or  subdivision  corresponding  to  the  num- 
ber of  divisions  or  subdivisions  of  the  natural  scale,  which  measures  the 
given  distance ;  and  the  line  drawn  along  the  sloped  edge  in  its  new  posi- 
tion will  be  the  line  required. 

The  natural  scale  may  be  used  advantageously  in  setting  off  the  dis- 
tances in  a  drawing,  and  the  corresponding  artificial  scale  in  drawing 
parallels  at  required  distances. 

The  advantages  of  Marquois's  scales  are  :  1st,  that  the  sight  is  greatly 
assisted  by  the  divisions  on  the  artificial  scale  being  so  much  larger  than 
those  of  the  natural  scale  to  which  the  drawing  is  constructed ;  2d,  that 
any  error  in  the  setting  of  the  index  produces  an  error  of  but 
one-third  the  amount  in  the  drawing. 

If  the  triangle  be  accurately  constructed,  these  scales  may 
be  advantageously  used  for  dividing  lines  with  accuracy  and 
despatch. 

Triangular  compasses. — Fig.  70  represents  this  instrument 
closed  up.  That  which  appears  in  the  fig.  as  one  limb  A,  con- 
sists of  a  pair  of  compasses  of  the  ordinary  construction.  The 
single  point  limb  B  has  a  compass  joint  at  <z,  by  which  its  point 
may  be  opened  at  right  angles  to  the  plane  of  the  pair  of  com- 
passes A,  when  the  three  points  will  form  a  triangle.  The  com- 
pass joint  a  is  firmly  attached  to  the  centre  of  the  compasses  A, 
which,  by  means  of  a  nut  and  screw  5,  may  be  turned  round 
without  moving  the  limbs,  of  which  it  is  the  centre.  The  double 
motion  thus  given  to  the  point  limb  B  (both  at  right  angles  to, 
and  parallel  to  the  plane  of  the.  compasses  A),  partakes  of  the 
nature  of  a  universal  joint,  and  enables  the  three  points  of  the 
instrument  to  be  placed  at  the  angular  points  of  any  shaped  tri- 
angle whatever.  This  instrument  is  chiefly  useful  in  transfer- 
ring of  points  from  one  paper  to  another.  The  two  points  of 
the  compasses  A  being  set  upon  such  points  of  the  drawing  as  have  been 
already  copied,  the  third,  B,  is  brought  upon  any  other  point ;  then,  by 
applying  the  points  A  to  the  corresponding  points  on  the  copy,  the  point 
B  will  establish  the  other  and  new  point  on  the  copy. 


28  DRAWING    INSTRUMENTS. 

Wholes  and  halves. — For  copying  and  reducing  drawing  to  half  size, 
compasses  called  wholes  arid  halves  are  used  (fig.  71),  in  which  the  longer 
legs  being  twice  the  length  of  the  shorter,  when  the  former  are  opened  to 
any  given  line,  the  shorter  ones  will  be  #' 
opened  to  the  half  of  that  line.  By  their 
means  then,  all  the  lines  of  a  drawing  may 
be  reduced  to  one-half,  or  enlarged  to 
double  their  length.  These  compasses  are 
also  useful  for  dividing  lines  by  continual 
bisections. 

The  proportional  compasses  (fig.  Y2)  are 
somewhat  similar  in  their  construction  to 
wholes  and  halves,  but  of  more  varied  ap- 
plication. The  principle  is  the  same,  with 
this  difference,  that  the  screw-joint  C 
passes  through  slides  moving  in  the  slots 
of  the  bars,  and  admits  of  the  centre  being 
rig.  71.  adjusted  for  various  relative  proportions 

between  the  openings  A  B  and  D  E.  Different  sets  of  num- 
bers are  engraved  on  the  outer  faces  of  the  bare,  and  by  these 
the  required  proportions  are  obtained.  The  instrument  must  Fi(r  72 
be  closed  for  adjustment,  and  the  nut  C  loosened  ;  the  slide  is  then  moved 
in  the  groove,  until  a  mark  across  it,  named  the  index,  coincides  with  the 
number  required  ;  which  done,  the  nut  is  tightened  again. 

The  scales  usually  engraved  on  these  compasses  are  named  Lines, 
Circles,  Planes,  and  Solids. 

The  scale  of  lines  is  numbered  from  1  to  10,  and  the  index  of  the  slide 
being  brought  to  any  one  of  these  divisions,  the  distance  D  E  will  measure 
A  B  in  that  proportion.  Thus,  if  the  index  be  set  to  4,  D  E  will  be  con- 
tained four  times  in  A  B. 

The  line  of  circles  extends  from  1  to  20,  and  the  index  being  set  to 
(say)  10,  D  E  will  be  the  tenth  part  of  the  circumference  of  the  circle, 
whose  radius  is  A  B. 

The  line  of  planes,  or  squares,  determines  the  proportion  of  similar 
areas.  Thus,  if  the  index  is  placed  at  3,  and  the  side  of  any  one  square  be 
taken  by  A  B  from  a  scale  of  equal  parts,  D  E  will  be  the  side  of  another 
square  of  one-third  the  area.  And  if  any  number  be  brought  to  the  index, 
and  the  same  number  be  taken  by  A  B  from  a  scale  of  equal  parts,  D  E 
will  be  the  square  root  of  that  number.  And  in  this  latter  case,  D  E  will 
also  be  a  mean  proportional  between  any  two  numbers,  whose  product  is 
equal  to  A  B, 


DRAWING   INSTRUMENTS.  29 

The  line  of  solids  expresses  the  proportion  between  cubes  and  spheres. 
Thus,  if  the  index  be  set  at  2,  and  the  diameter  of  a  sphere,  or  the  side  of 
a  cube,  be  taken  from  a  scale  of  equal  parts  by  A  B,  then  will  D  E  be  a 
diameter  or  side  of  a  sphere  or  cube  of  half  the  solidity.  And  if  the  slide 
be  set  to  (say)  8,  and  the  same  number  be  taken  from  a  scale  of  equal 
parts,  then  will  D  E  measure  2  on  the  same  scale,  or  the  cube  root  of  8. 

Beam  compasses  (fig.  73). — When  it  is  required  to  set  off  with  accuracy 
distances  of  considerable  extent,  or  describe  arcs  of  over  a  foot  radius,  the 
beam  compass  is  used.  This  instrument  consists  of  a  beam,  A  A,  of  any 
length  required,  generally  made  of  well-seasoned  mahogany ;  upon  its 


face  is  inlaid  throughout  its  whole  length  a  slip  of  holly  or  boxwood,  a  a, 
upon  which  are  engraved  the  divisions  or  scale,  either  feet  and  decimals  or 
inches  and  decimals,  or  whatever  particular  scale  may  be  required ;  but 
ordinary  beam  compasses  are  constructed  with  a  plain  beam,  with  no  scale 
whatever.  Two  brass  boxes,  B  and  C,  are  adapted  to  the  beam  ;  the  latter 
may  be  moved,  by  sliding,  to  any  part  of  its  length,  and  fixed  in  position 
by  tightening  the  clamp  screw  E.  Connected  with  the  brass  boxes  are 
the  two  points  of  the  instrument  G  and  H,  which  may  have  any  extent  of 
opening  by 'sliding  the  box  C  along  the  beam,  the  other  box,  B,  being 
firmly  fixed  at  one  extremity.  The  object  to  be  attained  ia  the  use  of  this 
instrument,  is  the  nice  adjustment  of  the  points  G  H  to  any  definite  dis- 
tance apart ;  this  is  accomplished  by  two  vernier  of  reading  plates  5  c,  each 
fixed  at  the  side  of  an  opening  in  the  brass  boxes  to  which  they  are  attached, 
and  afford  the  means  of  minutely  subdividing  the  principal  divisions  a  a 
on  the  beam,  which  appear  through  those  openings.  D  is  a  clamp  screw 
for  a  similar  purpose  as  the  screw  E,  namely,  to  fix  the  box  B,  and  prevent 
motion  in  the  point  it  carries  after  adjustment  to  position.  F  is  a  slow 
motion  screw,  by  which  the  point  G  may  be  moved  any  very  minute  quan- 
tity for  perfecting  the  setting  of  the  instrument,  after  it  has  been  set  as 
nearly  as  possible  by  the  hand  alone. 

The  method  of  setting  the  instrument  for  use  may  be  understood  from 
the  above  description  of  its  parts,  and  also  by  the  following  explanation  of 
the  method  of  examining  and  correcting  the  adjustment  of  the  vernier  J, 


30 


DRAWING   INSTRUMENTS. 


which  will  occasionally  get  deranged ;  this  verification  must  be  done  by 
means  of  a  detached  scale.  Tims,  suppose,  for  example,  that  our  beam 
compass  is  divided  to  feet,  inches,  and  tenths,  and  subdivided  by  the  ver- 
nier to  hundredths,  &c.  First  set  the  zero  division  of  the  vernier  to  the 
zero  of  the  principal  divisions  on  the  beam,  by  means  of  the  slow  motion 
screw  F.  This  must  be  done  very  nicely.  Then  slide  the  box  C,  with  its 
point  G,  till  the  zero  on  the  vernier  c  exactly  coincides  with  any  principal 
division  on  the  beam,  as  twelve  inches  or  six  inches,  &c.,  which  must  also 
be  done  very  accurately ;  then  apply  the  points  to  a  similar  detached 
scale,  and  if  the  adjustment  is  perfect,  the  interval  of  the  points  G  H  will 
measure  on  it  the  distance  to  which  they  were  set  on  the  beam.  If  they 
do  not  by  ever  so  small  a  quantity,  it  should  be  corrected  by  turning  the 
screw  F  till  the  points  exactly  measure  that  quantity  on  the  detached  scale ; 
then,  by  loosening  the  little  screws  which  hold  the  vernier  I  in  its  place, 
the  position  of  the  vernier  may  be  gradually  changed,  till  its  zero  coincides 
with  the  zero  on  the  beam,  and  then  tightening  the  screws  again,  the  ad- 
justment will  be  complete. 

Pat-table  or  turn-in  compasses  (fig.  74)  comprise  in  themselves  a  port- 
able case  of  drawing  instruments,  consisting  of  a  large  pair  of  compasses 

with  movable  points,  which  are 
also  so  contrived,  that  one  forms 
in  itself  a  small  pencil  bow,  the 
other  a  pen  bow  ;  and  when  the 
whole  instrument  is  put  together 
and  folded  up,  they  occupy  but  a 
space  three  inches  long,  and  may 
be  carried  in  the  pocket  without 
being  an  incumbrance. 

Fig.  74:  represents  the  instru- 
ment when  all  its  parts  are  to- 
gether. The  principal  legs  of  the 
instrument  are  F  and  G,  movable 
as  usual  by  a  joint  at  A.  The 
lower  joints,  B  and  C,  afford  the 
means  of  setting  the  point  limbs  D  and  E  perpendicular  to  the  paper. 

Each  of  the  point  limbs  may  be  removed  from  the  legs  F  and  G,  and 
by  means  of  their  joints  B  and  C,  form  perfect  instruments,  the  one  a  pen 
bow  represented  at  H,  and  the  other  a  pencil  bow,  shown  at  I K ;  the  point 
limbs  of  these  lesser  instruments  are  all  adapted  to  slide  into  the  principal 
legs  F  and  G  of  the  larger  one,  which  are  made  hollow  for  their  reception. 


Fig.  74. 


DRAWING    INSTRUMENTS. 


31 


It  may  easily  be  seen  from  the  engraving,  that  by  reversing  either  of 
the  points  in  the  principal  instrument,  it  may  be  supplied  with  a  pen  or 
pencil  as  may  be  required,  leaving  the  other  fine  or  plain  point  E  or  D  to 
act  as  a  centre. 

Mr.  Brunei  has  introduced  what  are  called  Tubular  Compasses,  in 
which  the  upper  part  of  the  legs  lengthens  out  like  the  slide  of  a  telescope, 
thus  giving  greater  extent  of  radius  when  required.  The  movable  legs  are 
double,  having  points  at  one  end,  and  a  pencil  or  pen  at  the  other ;  and 
they  move  on  pivots,  so  that  the  pen  or  pencil  can  be  instantly  substituted 
for  the  points,  or  vice  versa,  and  that  with  the  certainty  of  a  perfect  adjust- 
ment. The  design  is  very  ingenious,  and  offers  many  conveniences,  but 
the  instrument  is  too  delicate  for  ordinary  hands.  Without  extreme  care 
it  is  soon  disarranged. 

Large,  screw  dividers  (fig.  T5)  are  used  for  accurately  dividing  lines 
into  a  definite  number  of  equal  parts,  or  for  setting  off  equal  distances. 
A  is  the  centre  about  which  the  legs  A  0 
and  A  B  open  or  shut.  B  and  C  are 
joints,  by  which  the  point  limbs  may  be 
set  perpendicular  as  usual ;  the  extent  or 
opening  between  the  points  is  regulated 
by  a  screw  passing  through  a  socket 
F,  and  terminated  at  the  other  extremity 
by  a  milled  head  E,  by  which  the  screw 
is  turned  round.  Between  this  milled 
head  and  the  nearest  point  limb  is  fixed 
what  is  called  a  micrometer  head,  deci- 
mally divided  round  its  outer  or  cylindri- 
cal edge.  One  turn  of  the  screw  carries 
the  micrometer  head  completely  round  ; 
therefore,  when  part  of  a  turn  only  is 
given  to  the  screw,  the  divisions  on  the  head  show  what  fraction  of  a  turn 
has  been  given,  and  if  it  be  known  what  number  of  turns  or-threads  of  the 
screw  are  equal  to  one  inch,  the  points  of  these  compasses  may  be  thus  set 
to  any  small  definite  measure  of  length  with  the  utmost  precision.  The 
index  or  zero  for  reading  the  fraction  of  a  turn  of  the  screw  is  marked  on 
the  point  limb  below  J3.  Thus  this  instrument  may  be  considered  as  a 
beam  compass  of  small  dimensions  and  minute  accuracy. 

The  circular  protractor  (fig.  76)  is  one  of  the  best  kind  of  protractors. 
It  is  a  complete  circle,  A  A,  connected  with  its  centre  by  four  radii,  aaaa, 
The  centre  is  left  open,  and  surrounded  by  a  concentric  ring  or  collar,  5, 


Fig.  T5. 


32  DRAWING   INSTRUMENTS. 

which  carries  two  radial  bars,  c  c.  To  the  extremity  of  one  bar  is  a  pinion, 
d,  working  in  a  toothed  rack  quite  round  the  outer  circumference  of  the 
protractor.  To  the  opposite  extremity  of  the  other  bar,  <?,  is  fixed  a  vernier, 
which  subdivides  the  primary  divisions  on  the  protractor  to  single  minutes, 
and  by  estimation  to  30  seconds.  This  vernier  is  carried  round  the  pro- 
tractor by  turning  the  pinion  d.  Upon  each  radial  bar  c  c,  is  placed  a 
branch  e  e,  carrying  at  their  extremities  a  fine  steel  pricker,  whose  points 
are  kept  above  the  surface  of  the  paper  by  springs  placed  under  their  sup- 
ports, which  give  way  when  the  branches  are  pressed  downwards,  and 
allow  the  points  to  make  the  necessary  punctures  in  the  paper.  The 
branches  e  e  are  attached  to  the  bars  c  c,  with  a  joint  which  admits  of  their 
being  folded  backwards  over  the  instrument  when  not  in  use,  and  for  pack- 
ing in  its  case.  The  centre  of  the  instrument  is  represented  by  the  inter- 
section of  two  lines  drawn  at  right  angles  to  each  other  on  a  piece  of  plate 
glass,  which  enables  the  person  using  it  to  place  it,  so  that  the  centre,  or 
intersection  of  the  cross  lines,  may  coincide  with  any  given  point  on  the 
plan.  If  the  instrument  is  in  correct  order,  a  line  connecting  the  fine 
pricking  points  with  each  other  would  pass  through  the  centre  of  the  in- 
strument, as  denoted  by  the  before-mentioned  intersection  of  the  cross 
lines  upon  the  glass.  In  using  this  instrument,  the  vernier  should  first  be 
set  to  zero  (or  the  division  marked  360)  on  the  divided  limb,  and  then 


Fig.  76. 

placed  on  the  paper,  so  that  the  two  fine  steel  points  may  be  on  the  given 
line  (from  whence  other  and  angular  lines  are  to  be  drawn),  and  the  centre 
of  the  instrument  coincides  with  the  given  angular  point  on  such  line.  This 
done,  press  the  protractor  gently  down,  which  will  fix  it  in  position  by 
means  of  very  fine  points  on  the  under  side.  It  is  now  ready  to  lay  off  the 


DRAWING   INSTRUMENTS. 


33 


given  angle,  or  any  number  of  angles  that  may  be  required,  which  is  done 
by  turning  the  pinion  d  till  the  opposite  vernier  reads  the  required  angle. 
Then  press  downwards  the  branches  e  e,  which  will  cause  the  points  to 
make  punctures  in  the  paper  at  opposite  sides  of  the  circle  ;  which  being 
afterwards  connected,  the  line  will  pass  through  the  given  angular  point, 
if  the  instrument  was  first  correctly  set.  In  this  manner,  at  one  setting  of 
the  instrument,  a  great  number  of  angles  may  be  laid  off  from  the  same 
point. 

It  is  not  essential  that  the  centre  be  over  the  given  point,  when  applied 
to  the  given  line-,  provided  the  pricking  points  exactly  fall  upon  the  line, 
for  the  inclined  line  may  be  transferred  to  pass  through  the  given  angular 
point  by  a  parallel  ruler. 

The  pentagraph  (fig.  77)  is  used  for  the  copying  of  drawings  either  on 


Fig.  77. 

the  same  scale,  on  a  reduced  scale,  or  on  an  enlarged  scale,  as  may  be  re- 
quired.    It  is  represented  (fig.  77)  as  in  the  act  of  copying  a  plan  H,  upon 
3 


34  DRAWING   INSTRUMENTS. 

a  reduced  scale  h.  The  pentagraph  consists  of  four  rulers.  A,  B,  C,  and  D, 
made  of  stout  brass.  The  two  longer  rulers,  A  and  B,  are  connected  to- 
gether by,  and  have  a  motion  round  a  centre,  shown  at  the  upper  part  of 
the  engraving.  The  two  shorter  rulers  are,  in  like  manner,  connected  with 
each  other,  and  with  the  longer  rulers.  The  whole  instrument  is  supported 
by  small  pillars  resting  upon  ivory  castors,  a  a  a,  &c.,  which  have  a  motion 
in  all  directions.  The  rulers  A  and  C  have  each  an  equal  number  of  simi- 
lar divisions,  marked  £,  £,  &c. ;  and  likewise  a  sliding  index,  E  and  F, 
which  can  be  fixed  to  any  divisions  on  the  ruler  by  a  milled-headed  clamp 
screw  shown  in  the  engraving.  The  sliding  indexes,  E  and  F,  have  each 
of  them  a  tube  adapted  to  slide  on  a  pin,  rising  from  a  heavy  circular 
weight  called  the  fulcrum,  which  acts  as  a  centre  for  the  whole  instrument 
to  turn  upon  when  in  use,  or  to  receive  a  sliding  holder  with  a  pencil  or  a 
tracing  point,  as  may  be  required. 

To  explain  the  method  of  using  the  instrument,  the  engraving  repre- 
sents the  instrument  in  the  act  of  reducing  a  plan  to  a  scale  of  one  half  the 
original.  For  this  purpose  the  tracing  point  is  fixed  in  a  socket  at  G,  over 
the  original  drawing  H.  The  pencil  is  placed  in  a  similar  tube  or  socket  at 
F,  over  the  paper,  to  receive  the  copy ;  and  the  fulcrum  is  fixed  to  that  at  E, 
the  scale  being  one  half  the  original.  The  sliding  indices  were  first  clamped 
at  those  divisions  on  the  rulers  marked  |.  The  instrument  being  thus  set  for 
use,  if  correct,  the  three  points,  E,  F,  and  G,  will  be  in  one  straight  line, 
as  shown  by  the  dotted  line  in  the  figure.  This  will  invariably  be  the  case 
at  whatever  division  the  indices  may  be  set  to.  Now,  if  the  tracing  point 
G  be  passed  delicately  and  steadily  over  every  line  of  the  plan  H,  a  true 
copy,  but  of  one  half  the  scale  of  the  original,  will  be  marked  by  the  pencil 
at  F  on  the  paper  h  beneath  it.  The  fine  thread  represented  as  passing 
from  the  pencil  quite  round  the  further  extremity  of  the  instrument  to  the 
tracer  at  G,  is  to  enable  the  draftsman  at  the  tracing  point  to  raise  the 
pencil  from  the  paper,  whilst  he  passes  the  tracer  from  one  part  of  the  ori- 
ginal to  another,  and  prevents  false  lines  being  made  on  the  copy.  Like- 
wise, it  may  be  noticed,  that  the  pencil  holder  F  is  represented  as  sur- 
mounted by  a  cup,  which  is  for  the  purpose  of  putting  some  small  shot  in, 
to  press  the  pencil  heavier  upon  the  paper,  whenever  such  expedient  may 
be  found  necessary. 

If  the  object  had  been  to  enlarge  the  drawing  to  double  its  scale,  then 
the  tracer  must  have  been  placed  at  F,  and  the  pencil  at  G.  And  if  a  copy 
be  required,  retaining  the  scale  of  the  original,  then  the  slides  E  and  F 
must  be  placed  at  the  divisions  marked  1.  The  fulcrum  must  take  the 
middle  station,  and  the  pencil  and  tracer  those  on  the  exterior  rules  A  and 
B  of  the  instrument. 


DRAWING  INSTRUMENTS.  35 

The  camera  lucida  is  sometimes  used  for  copying  and  reducing  topo- 
graphical drawings.  A  description  of  the  use  of  this  instrument  will  be 
found  under  the  head  of  topographical  drawing. 

The  drawing  table  and  drawing  board. — The  usual  size  of  the  drawing 
table  should  be  from  5  to  6  feet  long,  and  3  feet  wide,  of  li  or  2  inch  white 
pine  plank  well  seasoned,  without  any  knots,  closely  joined,  glued,  dowelled, 
and  clamped.  It  should  be  fixed  on  a  strong  firm  frame  and  legs,  and  of 
such  a  height  that  the  draughtsman,  as  he  stands  up,  may  not  have  to 
stoop  to  his  work.  The  table  is  usually  provided  with  a  shallow  drawer  to 
hold  paper  or  drawings.  Drawing  tables  are  made  portable,  by  having 
two  horses  for  their  supports,  and  a  movable  drawing  board  for  the  top ; 
this  board  is  made  similar  to  the  top  of  the  drawing  table,  but  of  inch 
boards,  and  barred  at  the  ends.  Various  woods  are  used  for  the  purposes, 
but  white  pine  is  by  far  the  cheapest  and  best.  Drawing  boards  should  be 
made  truly  rectangular,  and  with  perfectly  straight  sides  for  the  use  of  the 
T  square.  Two  sizes  are  sufficient  for  common  purposes,  41  x  30  inches 
to  carry  double  elephant  paper  with  a  margin,  and  31  X  24  inches  for  im- 
perial and  smaller  sizes.  Boards  smaller  than  this  are  too  light  and  un- 
steady in  handling. 

Small  boards  are  occasionally  made,  as  loose  panels  fitting  into  a  frame, 
flush  on  the  drawing  surface,  with  buttons  on  the  back  to  secure  it  in 
position.  The  panel  is  mostly  of  white  pine,  with  a  hard-wood  frame. 


DRAWING-   PAPER. 

Drawing  paper,  properly  so  called,  is  made  to  certain  standard  sizes  as 
follows  :— 

Demy         .  v      ...  20  inches  by  15  i  inches. 

Medium,      '  .        .'  22£  "  ITi  " 

Eoyal,        .        ...  24  "  19  \  " 

Super  Eoyal,       ^_'.        .  2Ti  "  19i  " 

Imperial,    .         .       \  30  "  22  " 

Elephant,       .        ...  28  "  23  " 

Columbier,      1£        ".  35  "  23i  " 

Atlas,     .        .        ...  34  «  26  " 

Double  Elephant,       .  40  «  27  " 

Antiquarian,  .         .         .53  "  31  " 

Emperor,    ...  68  "  48  " 

Of  these,  Double  Elephant  is  the  largest  in  common  use  by  engineers,  and 
it  is  the  most  generally  useful  size  of  sheet.     Demy  and  Imperial  are  the 


36  DRAWING   INSTRUMENTS. 

only  other  sizes  worth  providing  for  a  drawing  establishment.  Whatman's 
white  paper  is  the  quality  most  usually  employed  for  finished  drawings ; 
it  will  bear  wetting  and  stretching  without  injury,  and  when  so  treated, 
receives  color  readily.  For  ordinary  working  drawings,  where  damp- 
stretching  is  dispensed  with,  cartridge  paper,  of  a  coarser,  harder,  and 
tougher  quality,  is  preferable.  It  bears  the  use  of  indiarubber  better,  re- 
ceives ink  on  the  original  undamped  surface  more  freely,  shows  a  fully 
better  line,  and  as  it  does  not  absorb  very  rapidly,  tinting  lies  better  and 
more  evenly  upon  it.  For  delicate  small-scale  line-drawing,  the  thick 
blue  paper,  such  as  is  used  for  ledgers,  &c.,  imperial  size,  answers  exceed- 
ingly well;  but  it  does  not  bear  damp-stretching  without  injury,  and 
should  be  merely  pinned  or  waxed  down  to  the  board.  "With  good  man- 
agement, there  is  no  ground  to  fear  the  shifting  of  the  paper.  Good  letter 
paper  receives  light  drawing  very  well ;  of  course  it  does  not  bear  much 
fatigue. 

Large  sheets,  destined  for  rough  usage  and  frequent  reference,  should 
be  mounted  on  linen,  previously  damped,  with  a  free  application  of  paste. 

Tracing  paper  is  a  preparation  of  tissue  paper,  transparent  and  quali- 
fied to  receive  ink  lines  and  tinting  without  spreading.  When  placed 
over  a  drawing  already  executed,  the  drawing  is  distinctly  visible  through 
the  paper,  and  may  be  copied  or  traced  directly  by  the  ink-instruments  ; 
thus  an  accurate  copy  may  be  made  with  great  expedition.  Tracings  may 
be  folded  and  stowed  away  very  conveniently  ;  but,  for  good  service,  they 
should  be  mounted  on  cloth,  or  on  paper  and  cloth,  with  paste. 

Tracing  paper  may  be  prepared  from  thick  tissue  paper,  by  sponging 
over  one  surface  with  a  mixture  of  one  part  raw  linseed  oil  and  five  spirits 
of  turpentine ;  five  gills  of  turpentine  and  one  of  oil  will  go  over  from 
forty  to  fifty  sheets  of  paper. 

Tracing  cloth  is  a  similar  preparation  of  linen,  and  is  preferable  for  its 
toughness  and  durability. 

Mouth  Glue,  for  the  sticking  of  the  edges  of  drawing  paper  to  the 
board,  is  made  of  glue  and  sugar  or  molasses ;  it  melts  at  the  temperature 
of  the  mouth,  and  is  convenient  for  the  draughtsman. 

Drawing  paper  may  be  fixed  down  on  the  drawing  board  by  the  pins 
at  the  corners,  by  weights,  or  by  gluing  the  edges.  The  first  is  sufficient 
when  no  shading  or  coloring  is  to  be  applied,  and  if  the  sheet  is  not  to  be 
a  very  long  time  on  the  board ;  and  it  has  the  advantage  of  preserving 
the  paper  in  its  natural  state.  For  shaded  or  tinted  drawings,  the  paper 
must  be  damped  and  glued  at  the  edges,  as  the  partial  wetting  of  paper, 
loose  or  fixed  at  the  corners  merelyj  by  the  water  colors  distorts  the  surface. 


DRAWING   INSTRUMENTS.  37 

Damp-stretching  is  done  as  follows : — The  edges  of  the  paper  should 
first  be  cut  straight,  and,  as  near  as  possible,  at  right  angles  with  each 
other ;  also  the  sheet  should  be  so  much  larger  than  the  intended  drawing 
and  its  margin,  as  to  admit  of  being  afterwards  cut  from  the  board,  leaving 
the  border  by  which  it  is  attached  thereto  by  glue  or  paste,  as  we  shall 
next  explain. 

The  paper  must  first  be  thoroughly  and  equally  damped  with  a  sponge 
and  clean  water,  on  the  opposite  side  from  that  on  which  the  drawing  is  to 
be  made.  When  the  paper  absorbs  the  water,  which  may  be  seen  by  the 
wetted  side  becoming  dim,  as  its  surface  is  viewed  slantwise  against  the  light, 
it  is  to  be  laid  on  the  drawing  board  with  the  wetted  side  downwards,  and 
placed  so  that  its  edges  may  be  nearly  parallel  with  those  of  the  board ; 
otherwise,  in  using  a  T  square,  an  inconvenience  may  be  experienced. 
This  done,  lay  a  straight  flat  ruler  on  the  paper,  with  its  edge  parallel  to, 
and  about  half  an  inch  from  one  of  its  edges.  The  ruler  must  now  be  held 
firm,  while  the  said  projecting  half  inch  of  paper  be  turned  up  along  its 
edge  ;  then,  a  piece  of  solid  or  mouth  glue,  having  its  edge  partially  dis- 
solved by  holding  it  in  boiling  or  warm  water  for  a  few  seconds,  must  be 
passed  once  or  twice  along  the  turned  up  edge  of  the  paper,  after  which, 
by  sliding  the  ruler  over  the  glued  border,  it  will  be  again  laid  flat,  and 
the  rule  being  pressed  down  upon  it,  that  edge  of  the  paper  will  adhere  to 
the  board.  If  sufficient  glue  has  been  applied,  the  ruler  may  be  removed 
directly,  and  the  edge  finally  rubbed  down  by  an  ivory  book-knife,  or  by 
the  bows  of  a  common  key,  by  rubbing  on  a  slip  of  paper  placed  on  the 
drawing  paper,  so  that  the  surface  of  the  latter  may  not  be  soiled,  which 
will  then  firmly  cement  the  paper  to  the  board.  This  done,  another  but 
adjoining  edge  of  the  paper  must  be  acted  upon  in  like  manner,  and  then 
the  remaining  edges  in  succession ;  we  say  the  adjoining  edges,  because 
we  have  occasionally  observed,  that  when  the  opposite  and  parallel  edges 
have  been  laid  down  first,  without  continuing  the  process  progressively 
round  the  board,  a  greater  degree  of  care  is  required  to  prevent  undula- 
tions in  the  paper  as  it  dries. 

Sometimes  strong  paste  is  used  instead  of  glue ;  but  as  this  takes  a 
longer  time  to  set,  it  is  usual  to  wet  the  paper  also  on  the  upper  surface  to 
within  an  inch  of  the  paste  mark,  care  being  taken  not  to  rub  or  injure  the 
surface  in  the  process.  The  wetting  of  the  paper  in  either  case  is  done  for 
the  purpose  of  expanding  it ;  and  the  edges  being  fixed  to  the  board  in  its 
enlarged  state,  act  as  stretchers  upon  the  paper,  while  it  contracts  in  dry- 
ing, which  it  should  be  allowed  to  do  gradually.  All  creases  or  undula- 


38  DRAWING   INSTRUMENTS. 

tions  by  this  means  disappear  from  the  surface,  and  it  forms  a  smooth 
plane  to  receive  the  drawing. 

To  remove  the  paper  after  the  drawing  is  finished,  cut  off  inside  the 
pasted  edge,  and  remove  the  edge  by  warm  water  and  the  knife. 

With  panelled  boards,  the  panel  is  taken  out,  and  the  frame  inverted  ; 
the  paper,  being  first  damped  on  the  back  with  a  sponge  slightly  charged 
with  water,  is  applied  equally  over  the  opening  to  leave  equal  margins, 
and  is  pressed  and  secured  into  its  seat  by  the  panel  and  bars. 


MOUNTING  PAPER  AND  DRAWINGS,   VARNISHING,   ETC. 

In  mounting  paper  upon  canvas,  the  latter  should  be  well  stretched 
upon  a  smooth  flat  surface,  being  damped  for  that  purpose,  and  its  edges 
glued  down,  as  was  recommended  in  stretching  drawing  paper.  Then  with 
a  brush  spread  strong  paste  upon  the  canvas,  beating  it  in  till  the  grain  of 
the  canvas  be  all  filled  up  ;  for  this,  when  dry,  will  prevent  the  canvas 
from  shrinking  when  subsequently  removed ;  then,  having  cut  the  edges 
of  the  paper  straight,  paste  one  side  of  every  sheet,  and  lay  them  upon  the 
canvas  sheet  by  sheet,  overlapping  each  6ther  a  small  quantity.  If  the 
drawing  paper  is  strong,  it  is  best  to  let  every  sheet  lie  five  or  six  minutes 
after  the  paste  is  put  on  it,  for  as  the  paste  soaks  in,  the  paper  will  stretch, 
and  may  be  better  spread  smooth  upon  the  canvas  ;  whereas,  if  it  be  laid 
on  before  the  paste  has  moistened  the  paper,  it  will  stretch  afterwards  and 
rise  in  blisters  when  laid  upon  the  canvas.  The  paper  should  not  be  cut 
off  from  its  extended  position  till  thoroughly  dry,  which  should  not  be 
hastened,  but  left  in  a  diy  room  to  do  so  gradually,  if  time  permit ;  if  not, 
it  may  be  exposed  to  the  sun,  unless  in  the  winter  season,  when  the  help 
of  a  fire  is  necessary,  provided  it  is  not  placed  too  near  a  scorching  heat. 

In  joining  two  sheets  of  paper  together  by  overlapping,  it  is  necessary, 
in  order  to  make  a  neat  joint,  to  feather  edge  each  sheet ;  this  is  done  by 
carefully  cutting  with  a  knife  half  way  through  the  paper  near  the  edges, 
and  on  the  sides  which  are  to  overlap  each  other  ;  then  strip  off  a  feather- 
edged  slip  from  each,  which,  if  done  dexterously,  will  form  a  very  neat 
and  efficient  joint  when  put  together. 

For  mounting  and  varnishing  drawings  or  prints,  stretch  a  piece  of  linen 
on  a  frame,  to  which  give  a  coat  of  isinglass  or  common  size,  paste  the  back 
of  drawing,  which  leave  to  soak,  and  then  lay  it  on  the  linen.  When  dry, 
give  it  at  least  four  coats  of  well  made  isinglass  size,  allowing  it  to  dry  be- 


DRAWING   INSTRUMENTS.  39 

tween  each  coat.     Take  Canada  balsam  diluted  with  the  best  oil  of  turpen- 
tine, and  with  a  clean  brush  give  it  a  full  flowing  coat. 


MANAGEMENT   OF  THE   INSTRUMENTS. 

In  constructing  preparatory  pencil-drawings,  it  is  advisable,  as  a  rule 
of  general  application,  to  make  no  more  lines  upon  the  paper  than  are  ne- 
cessary to  the  completion  of  the  drawing  in  ink ;  and  also  to  make  these 
lines  just  so  dark  as  is  consistent  with  the  distinctness  of  the  work.  With 
respect  to  the  first  idea,  it  is  of  frequent  application :  in  the  case,  for 
example,  of  the  teeth  of  spur  wheels,  where,  in  many  instances,  all  that  is 
necessary  to  the  drawing  of  their  end  view  in  ink  are  three  circles,  one 
of  them  for  the  pitch  line,  and  the  two  others  for  the  tops  and  bottoms 
of  the  teeth  ;  and  again,  to  draw  the  face  view  of  the  teeth,  that  is,  in  the 
edge  view  of  the  wheel,  we  have  only  to  mark  off  by  dividers  the  positions 
of  the  lines  which  compose  the  teeth,  and  draw  four  pencil  lines  for  the 
two  sides,  and  the  top  and  bottom  of  the  elevation.  And  here  we  may 
remark  the  inconvenience  of  that  arbitrary  rule,  by  which  it  is  by  some  in- 
sisted that  the  pupil  should  lay  down  in  pencil  every  line  that  is  to  be 
drawn,  before  finishing  it  in  ink.  It  is  often  beneficial  to  ink  in  one  part 
of  a  drawing,  before  touching  other  parts  at  all ;  it  prevents  confusion, 
makes  the  first  part  of  easy  reference,  and  allows  of  its  being  better  done, 
as  the  surface  of  the  paper  inevitably  contracts  dust,  and  becomes  other- 
wise soiled  in  the  course  of  time,  and  therefore  the  sooner  it  is  done  with 
the  better. 

Circles  and  circular  arcs  should,  in  general,  be  inked  in  before  straight 
lines,  as  the  latter  may  be  more  readily  drawn  to  join  the  former,  than  the 
former  the  latter.  When  a  number  of  circles  are  to  be  described  from  one 
centre,  the  smaller  should  be  inked  first,  while  the  centre  is  in  better  con- 
dition. When  a  centre  is  required  to  bear  some  fatigue,  it  should  be  pro- 
tected with  a  thickness  of  stout  card  glued  or  pasted  over  it,  to  receive  the 
compass-leg. 

Indiarubber  is  the  ordinary  medium  for  cleaning  a  drawing,  and  for  cor- 
recting errors  in  the  pencil.  For  slight  work  it  is  quite  suitable ;  that  sub- 
stance, however,  operates  to  destroy  the  surface  of  the  paper  ;  and  by  re- 
peated application,  it  so  ruffles  the  surface,  and  imparts  an  unctuosity  to  it, 
as  to  spoil  it  for  fine  drawing,  especially  if  ink  shading  or  coloring  is  to  be  ap- 
plied. It  is  much  better  to  leave  trivial  errors  alone,  if  corrections  by  the 
pencil  may  be  made  alongside  without  confusion ;  as  it  is,  in  such  a  case, 
time  enough  to  clear  away  superfluous  lines  when  the  inking  is  finished. 


40  DRAWING  INSTRUMENTS. 

For  cleaning  a  drawing,  a  piece  of  bread  two  days  old  is  preferable  to 
indiarubber,  as  it  cleans  the  surface  well  and  does  not  injure  it.  "When  ink 
lines  to  any  considerable  extent  have  to  be  erased,  a  small  piece  of  damped 
soft  sponge  may  be  rubbed  over  them  till  they  disappear.  As,  however, 
this  process  is  apt  to  discolor  the  paper,  the  sponge  must  be  passed  through 
clean  water,  and  applied  again  to  take  up  the  straggling  ink.  For  ordi- 
nary small  erasures  of  ink  lines,  a  sharp  rounded  pen-blade  applied  lightly 
and  rapidly  does  well,  and  the  surface  may  be  smoothed  down  by  the 
thumb-nail.  In  ordinary  working  drawings,  a  line  may  readily  be  taken 
out  by  damping  it  with  a  hair  pencil,  and  quickly  applying  the  indiarub- 
ber ;  and  to  smooth  the  surface  so  roughened,  a  light  application  of  the 
knife  is  expedient.  In  drawings  intended  to  be  highly  finished,  particular 
pains  should  be  taken  to  avoid  the  necessity  for  corrections,  as  every  thing 
of  this  kind  detracts  from  the  appearance. 

In  using  the  square,  the  more  convenient  way  is  to  draw  the  lines  off 
the  left  edge  with  the  right  hand,  holding  the  stock  steadily  but  not  very 
tightly  against  the  edge  of  the  board  with  the  left  hand.  The  convenience 
of  the  left  edge  for  drawing  by  is  obvious,  as  we  are  able  to  use  the  arms 
more  freely,  and  we  see  exactly  what  we  are  doing. 

To  draw  lines  in  ink  with  the  least  amount  of  trouble  to  himself,  the 
mechanical  draughtsman  ought  to  take  the  greater  amount  of  trouble  with 
his  tools.  If  they  be  well  made,  and  of  good  stuff  originally,  they  ought 
to  last  through  three  generations  of  draughtsmen ;  their  working  parts 
should  be  carefully  preserved  from  injury,  they  should  be  kept  well  set, 
and,  above  all,  scrupulously  clean.  The  setting  of  instruments  is  a  matter 
of  some  nicety,  for  which  purpose  a  small  oil-stone  is  convenient.  To  dress 
up  the  tips  of  the  blades  of  the  pen  or  of  the  bows,  as  they  are  usually 
worn  unequally  by  the  customary  usage,  they  may  be  screwed  up  into  con- 
tact in  the  first  place,  and  passed  along  the  stone,  turning  upon  the  point 
in  a  directly  perpendicular  plane,  till  they  acquire  an  identical  profile. 
Being  next  unscrewed  and  examined  to  ascertain  the  parts  of  unequal 
thickness  round  the  nib,  the  blades  are  laid  separately  upon  their  backs  on 
the  stone,  and  rubbed  down  at  the  points,  till  they  be  brought  up  to  an 
edge  of  uniform  fineness.  It  is  well  to  screw  them  together  again,  and  to 
pass  them  over  the  stone  once  or  twice  more,  to  bring  up  any  fault ;  to  re- 
touch them  also  on  the  outer  and  inner  side  of  each  blade,  to  remove  barbs 
or  frasing  ;  and,  finally,  to  draw  them  across  the  palm  of  the  hand. 

The  China  ink,  which  is  commonly  used  for  line-drawing,  ought  to  be 
rubbed  down  in  water  to  a  certain  degree,  avoiding  the  sloppy  aspect  of 
light  lining  in  drawings,  and  making  the  ink  just  so  thick  as  to  run  freely 


DRAWING   INSTRUMENTS.  41 

from  the  pen.  This  medium  degree  may  be  judged  of  after  a  little  practice 
by  the  appearance  of  the  ink  on  the  pallet.  The  best  quality  of  ink  has  a 
soft  feel  when  wetted  and  smoothed;  free  from  grit  or  sediment,  and 
musky.  The  rubbing  of  China  ink  in  water  tends  to  crack  and  break  away 
the  surface  at  the  point ;  this  may  be  prevented  by  shifting  at  intervals  the 
position  of  the  stick  in  the  hand  while  being  rubbed,  and  thus  rounding  the 
surface.  ]S"or  is  it  advisable,  for  the  same  reason,  to  bear  very  hard,  as 
the  mixture  is  otherwise  more  evenly  made,  and  the  enamel  of  the  pallet 
is  less  rapidly  worn  off.  When  the  ink,  on  being  nibbed  down,  is  likely 
to  be  for  some  time  required,  a  considerable  quantity  of  it  should  be  pre- 
pared, as ,  the  water  continually  vaporises ;  it  will  thus  continue  for  a 
longer  time  in  a  condition  fit  for  application.  The  pen  should  be  levelled 
in  the  ink,  to  take  up  a  sufficient  charge  ;  and  to  induce  the  ink  to  enter 
the  pen  freely,  the  blades  should  be  lightly  breathed  upon  before  immer- 
sion. After  each  application  of  ink,  the  outsides  of  the  blades  should  be 
cleaned,  to  prevent  any  deposit  of  ink  upon  the  edge  of  the  squares. 

To  keep  the  blades  of  his  inkers  clean,  is  the  first  duty  of  a  draughtsman 
who  is  to  make  a  good  piece  of  work.  Pieces  of  blotting  or  unsized  paper 
and  cotton  velvet,  washleather,  or  even  the  sleeve  of  a  coat,  should  always 
be  at  hand  while  a  drawing  is  being  inked.  "When  a  small  piece  of  blot- 
ting paper  is  folded  twice  so  as  to  present  a  corner,  it  may  usefully  be 
passed  between  the  blades  of  the  pen  now  and  then,  as  the  ink  is  liable  to 
deposit  at  the  point  and  obstruct  the  passage,  particularly  in  fine  lining ; 
and  for  this  purpose  the  pen  must  be  unscrewed  to  admit  the  paper.  But 
this  process' may  be  delayed  by  drawing  the  point  of  the  pen  over  a  piece 
of  velvet,  or  even  over  the  surface  of  thick  blotting  paper  ;  either  method 
clears  the  point  for  a  time.  As  soon  as  any  obstruction  takes  place,  the 
pen  should  be  immediately  cleaned,  as  the  trouble  thus  taken  will  always 
improve  and  expedite  the  work.  If  the  pen  should  be  laid  down  for  a  short 
time  with  the  ink  in  it,  it  should  be  unscrewed  to  keep  the  points  apart, 
and  so  prevent  deposit ;  and  when  done  with  altogether  for  the  occasion, 
it  ought  to  be  thoroughly  cleaned  at  the  nibs.  This  will  preserve  its  edges 
and  prevent  rusting. 

For  the  designing  of  machinery,  it  is  very  convenient  to  have  some 
scale  of  reference  by  which  to  proportion  the  parts ;  for  this  purpose,  a 
vertical  and  horizontal  scale  may  be  drawn  on  the  walls  of  the  room. 


42  GEOMETRICAL   PROBLEMS. 


GEOMETEICAL  PEOBLEMS. 

ON   STRAIGHT  LINES. 

IT  is  desirable  that  the  beginner  should  construct  the  following  prob- 
lems, not  copying  them,  but  adopting  some  scale,  which  will  give  him  the 
use  of  the  scale,  and  imprint  the  problems  on  his  memory. 
PROBLEM  I. — To  draw  a  straight  line  through  given  points. 
Let  A  and  B  (fig.  78)  be  two  given  points,  represented  by  the  intersec- 
tion of  two  lines,  or  pricked 

I 4 . — — 5 — : — |      into  the  surface.     Surround 

the  points  by  small  circles, 

Fig.  T8. 

when  advisable  for  assisting 

to  define  their  locality,  as  thus  0  ;  place  the  straight  edge  at  or  so  near 
the  points,  that  the  point  of  the  pen  or  pencil  may  pass  through  them,  and 
draw  the  line  firmly  and  steadily. 

Lines  in  drawing  are  divided  into  several  classes,  asfutt,  broken,  dotted, 
and  broken  and  dotted,  &c. ;  these  again  are  divided  into  fine,  medium,  and 
heavy,  according  to  the  breadth  of  the  line  (fig.  79). 

The  lines  of  a  problem  which  are  either  given  or  are  to  be  found,  and 

Full.  Broken.  Dotted.  Broken  and  dotted. 


Fig.  79. 

the  outlines  of  an  object  that  can  be  seen  from  the  point  of  view  in  which 
it  is  represented  should  be  full,  and  either  fine,  medium,  or  heavy,  accord- 
ing to  the  particular  efifect  that  the  draughtsman  wishes  to  give.  The  por- 
tions of  the  outline  that  are  hidden  from  view,  but  which  are  requisite  to 
give  a  complete  idea  of  the  object,  should  be  dotted  or  broken. 


GEOMETRICAL   PROBLEMS.  43 

The  other  lines  are  used  for  conventional  purposes  by  the  draughtsman, 
as  boundaries  of  parishes  or  estates,  or  to  show  a  change  in  position  of  an 
object,  &c.,  &c. 

PKOB.  II. — To  set  off  a  gi/ven  distance  along  a  straight  line  C  D,froin 
a  given  point  A  on  it  (fig.  80). 

Take  off  the  given  distance  from  the  scale  of  equal  parts  with  the 
dividers.  Set  one  foot  of  the 

dividers  on  A,  and  bring  the    c?     .  A* %  s  p 

other  foot  upon  the  line,  and  Fig.  so. 

mark  the  point  B,  either  by 

pricking  with  the  foot  of  the  dividers,  or  by  a  small  dot  with  the  sharp 

point  of  a  lead  pencil. 

When  the  distance  as  at  A  to  be  set  off  is  too  small  to  be  taken  off 
from  the  scale  with  accuracy,  set  off  any  convenient  distance  A  5  greater 
than  the  given  distance  ;  set  off  from  ~b  towards  A,  the  length  by  which 
A  b  is  greater  than  the  given  distance  ;  the  part  A  a  will  be  the  required 
distance. 

To  set  off  a  number  of  distances  on  a  straight  line,  set  off  their  succes- 
sive sums.  Thus,  to  set  off  successively  the  distances  =  10,  15,  20,  set  off 
as  above  10,  25,  and  45,  always  starting  from  the  same  point.  The  object 
of  performing  the  operation  in  this  manner  is  to  avoid  carrying  forward 
any  inaccuracy  that  might  be  made  were  the  respective  distances  set  off 
separately,  one  after  the  other.  If  the  distances  to  be  set  off  are  equal,  it 
will  be  more  accurate  to  set  off  a  distance  equal  either  to  the  whole  aggre- 
gate, or  such  a  number  of  them  as  can  be  contained  by  the  compasses,  and 
then  dividing  the  line  into  the  required  parts. 

PKOB.  III. — To  divide  a  given  line  into  two  equal  parts  (fig.  81). 

Open  the  dividers  to  as  near  as  possible  half  the  given  line,  place  one 
point  of  the  dividers  on  the  end  of  the  line,  bring  the  other  point  to  the 
line,  and  turn  on  this  point ;  if  now  the  point  of  the  dividers  coincide  with 


Fig.  81. 


the  other  end  of  the  line,  we  have  the  division  required ;  but  should  the  point 
of  the  dividers  fall  within  or  without  the  end  of  the  line,  divide  this  deficit 
or  excess  by  the  eye  into  two  equal  parts,  and  contract  or  open  the  dividers 


44  GEOMETRICAL  PROBLEMS. 

to  this  point,  and  apply  them  again  as  at  first ;  perform  the  operation  till 
the  revolution  of  the  compasses  coincides  with  the  given  line.  Thus  (fig. 
81),  suppose  it  were  required  to  divide  the  line  A  B  into  two  equal  parts, 
and  the  distance  A  C'  was  the  first  guess  or  opening  of  the  dividers ;  turn- 
ing on  the  point  C',  the  point  of  the  dividers  that  was  at  A  falls  on  the 
point  D  beyond  B,  keeping  the  point  of  the  dividers  still  on  the  point  C', 
open  them  till  they  embrace  the  distance  C'  J,  5  being  at  or  near  as  can 
be  judged  by  the  eye  the  centre  of  D  B ;  begin  again  from  the  point  A  with 
the  distance  C'  5  contained  in  the  dividers,  and  apply  the  distance  as  at 
first,  dividing  the  deficit  or  excess  of  the  two  revolutions,  till  the  point  of 
the  dividers  that  was  at  A  falls  by  revolution  on  B.  The  eye,  by  practice, 
becomes  so  accustomed  to  this  means  of  division,  that  a  plan  may  be  re- 
duced to  half  scale  as  quickly  and  with  as  little  chance  of  mistake  as  by  the 
proportional  compasses. 

To  divide  a  line  into  any  number  of.  equal  parts. — If  the  number  is 
divisible  by  two,  bisect  the  line,  or  divide  it  into  two  equal  parts  as  above, 
and  continue  this  as  long  as  the  number  remaining  is  divisible  by  two  ;  but 
when  the  number  is  uneven,  measure  the  given  line  on  the  scale,  divide 
numerically  the  length  thus  found  by  the  number  of  parts  into  which  it 
is  to  be  divided,  and  take  on  the  scale  as  accurately  as  possible  the  quotient 
thus  obtained ;  apply  this  length  successively  on  the  line,  and  if  the  last 
distance  set  off  does  not  agree  with  the  extremity  of  the  line  ;  thus  if,  as  in 
fig.  81,  when  the  line  is  to  be  divided  into  two  parts,  the  repeated  length 
exceeds  the  line,  divide  this  excess  by  the  eye  into  as  many  parts  as  the 
given  line  is  to  be  divided,  and  close  the  dividers  so  as  to  include  a 
length  less  by  one  of  these  parts.  If  the  point  D  should  fall  inside 
\j,/  B,  divide  the  deficit  as  before  by 

the  number  of  parts,   but  open 
\  the    dividers    by  one    of  these 

parts. 

The  above  problems  may  be 
constructed  geometrically  as  fol- 
lows : — To  Insect  or  divide  into 
two  equal  parts  a  given  line  A  B. 
/  From  A  and  B,  with  any  radius 

greater  than  half  of  A  B,  describe 
Fig- 82<  arcs  intersecting  each  other  above 

and  below  the  given  line,  the  line  C  D  connecting  these  intersections  will 
bisect  A  B,  and  also  be  perpendicular  to  it. 


GEOMETRICAL   PROBLEMS. 


45 


PROB.  IY. — To  divide  a  given  line  into  a  given  number  of  equal  parts. 

Let  A  B  be  the  distance 
to  be  divided,  for  example, 
into  four  equal  parts ;  draw 
the  line  A  C,  making  an 
acute  angle  with  A  B  ;  on 
A  C  lay  off  any  four  equal 
distances,  each  as  near  as 
may  be  to  1  of  A  B,  con- 
nect the  last  division  4  with 
B,  and  through  the  other 
points  1,  2,  3,  draw  lines 
parallel  to  4  B,  the  inter- 
sections of  these  lines  5,c,<7, 
with  the  line  A  B,  will  divide  it  into  four  equal  parts. 

PROB.  Y. — To  draw  a  perpendicular  to  a  straight  line,  from,  a,  point 
without  it. 

\st  Method  (fig.  84). — From  the  point  A,  with  a^ sufficient  radius,  cut 
the  given  line  at  F  and  G,  and  from  these  points  describe  arcs  cutting  at  E  ; 
through  E  draw  A  E,  which  will  be  the  perpendicular  required.  If  there  be 
no  room  below  the  line,  the  intersection  may  be  taken  above  ;  that  is,  be- 
tween the  line  and  the  given  point.  This  mode  is  not,  however,  likely  to 
be  as  exact  in  practice  as  the  one  given. 


Fig.  83. 


Fig.  84 


Fig.  85. 


2<#  Method  (fig.  85). — From  any  two  points  B  and  C,  at  some  distance 
apart,  in  the  given  line,  and  with  radii  B  A,  C  A,  respectively,  describe  arcs 


GEOMETRICAL   PROBLEMS. 


cutting  at  A  and  D.  Draw  the  perpendicular  required,  A  D.  This  method 
is  useful  where  the  given  point  is  opposite  the  end  of  the  line,  or  nearly  so. 

PROB.  YI. — To  draw  a  perpendicular  to  a  straight  line  from,  a  given 
point  A  in  that  line. 

1st  Method  (fig.  86). — With  any  radius,  from  the  given  point  A,  in  the 
given  line  B  C,  cut  the  line  at  B  and  C ;  with  a  greater  radius  describe  arcs 
from  B  and  C,  cutting  each  other  at  D,  and  draw  D  A  the  perpendicular. 

E/' 


\F 


A 

Fig.  66. 


Fig.8T. 


2d  Method  (fig.  87).— From  any  centre  F,  above  B  C,  describe  a  circle 
passing  through  the  given  point  A,  and  cutting  the  given  line  at  D  ;  draw 
D  F,  and  produce  it  to  cut  the  circle  at  E  ;  and  draw  A  E  the  perpendicular. 
This  method  is  useful  when  the  point  A  is  at  or  near  one  end  ;  and  in  prac- 
tice, it  is  expedient  in  the  first  place  to  strike  out  a  preliminary  arc,  of  any 
convenient  radius,  from  the  point  A,  as  any  point  in  that  arc  may  be 
chosen  for  the  centre  F,  with  the  certainty  that  the  arc  from  this  centre 
will  pass  through  A,  without  the  delay  of  adjusting  the  point  of  the  com- 
pass to  it.  This  expedient  is  of  general  use  where  an  arc  is  to  be  passed 
through  a  given  point,  and  particularly  if  the  point  of  the  pencil  be  round 
or  misshapen,  and  therefore  uncertain. 

3d  Method  (fig.  88).— From  A  describe  an  arc  E  C,  and  from  E,  with  the 


Fig.  69. 


<*ame  radius,  the  arc  A  C,  cutting  the  other  at  C ;  through  C  draw  a  line  E  C  D, 
and  set  off  C  D  equal  to  C  E,  and  through  D  draw  A  D  the  perpendicular 


GEOMETRICAL  PROBLEMS.  47 

required.  This  method,  like  the  previous  one,  is  useful  when  the  point  A 
is  at  one  end. 

±th  Method  (fig.  89). — From  the  given  point  A,  set  off  a  distance  A  E 
equal  to  three  parts,  by  any  scale ;  and  on  the  centres  A  and  E,  with  radii 
of  four  and  five  parts  respectively,  describe  arcs  intersecting  at  C.  Draw 
A  C  for  the  perpendicular  required.  This  method  is  most  useful  on  very  large 
scales,  where  straight  edges  are  inapplicable,  as  in  laying  down  perpendi- 
culars or  right  angles  on  the  ground ;  as  in  laying  out  the  corners  of  houses, 
beams  and  girders  may  be  set  square  with  the  sides  of  the  houses,  columns 
and  the  like  may  be  set  perpendicularly  by  the  same  method.  The 
numbers  3,  4,  5,  are,  it  is  to  be  observed,  taken  to  measure  respectively — 
the  base,  the  perpendicular,  and  the  slant  side  of  the  triangle  AEG.  Any 
multiples  of  these  numbers  may  be  used  with  equal  propriety,  as  6,  8,  10, 
or  9,  12,  15,  whether  feet,  yards,  or  any  other  measure  of  length. 

PROB.  YII. — To  draw  a  straight  line  parallel  to  a  given  line,  at  a  given 
distance  apart  (fig.  90).  „ 


From  the  centres  A,  B,  in  the  given  line,    ""' 
with  the  given  distance  as  radius,  describe 
arcs  C,  D  ;  and  draw  the  parallel  line  C  D 
touching  the  arcs.     The  method  of  drawing 
tangents  will  be  afterwards  shown ;   mean-          -^  •" 

time,  in  all  ordinary  cases,  the  line  C  D  may 
be  drawn  by  simply  applying  a  straight  edge  by  the  eye. 

PROB.  YIII. — To  draw  a  parallel  through  a  given  point. 

1st  Method  (fig.  91). — With  a  radius  equal  to  the  distance  of  the  given 
point  C  from  the  given  line  A  B,  describe  the  arc  D  from  B,  taken  con- 
siderably distant  from  C  ;  draw  the  parallel  through  G  to  touch  the  arc  D. 


A/  In 


/f 

Fig.  91.  Fig.  92. 

%d  Method  (fig.  92).— From  A,  the  given  point,  describe  the  arc  F  D, 
cutting  the  given  line  at  F ;  from  F,  with  the  same  radius,  describe  the 
arc  E  A ;  and  set  off  F  D  equal  to  E  A.  Draw  the  parallel  through 
the  points  A,  D. 


GEOMETRICAL   PROBLEMS. 


PROB.  IX.  —  To  construct  an  angle 
equal  to  a  given  angle  (fig.  93). 

Thus,  on  the  line  a  5  to  construct  an  an- 
gle which  shall  be  equal  to  the  given  angle 
CAB.  With  the  dividers  describe  the  arc 
C  B ;  from  the  point  #,  with  the  same  ra- 
dius describe  c  1> ;  with  the  dividers  mea- 
sure the  length  of  the  arc  C  B,  and  on  c  & 
lay  off  this  distance ;  through  c  draw  c  a, 
and  we  have  the  required  angle  or  open- 
ing c  a  5,  equal  to  the  given  angle  CAB. 
PKOB.X. — From  a  point  A  of  a  given 
line  D  E,  to  draw  a  line  making  an  angle 
of  60°  with  the  given  line  (fig.  94). 

Take  any  convenient  distance  in  the 
dividers,  and  from  A  describe  the  arc  B  C.  From  B,  with  the  same  dis- 
tance, describe  an  arc,  and  mark  the  point  C  where  the  arcs  cross.  Draw 
the  line  A,  C.  This  line  will  make  with  the  given  one  the  required 
angle  of  60°. 


Fig.  93. 


^ 


Fig.  94 


Fig.  95. 


PKOB.  XI. — From  a  point  'Bon  a  given  line  DE,  to  draw  a  line  making 
an  angle  of  45°  with  it  (fig.  95). 

Set  off  any  distance  B  a,  along  D  E,  from  B.  Construct  a  perpendicu- 
lar to  D  E  at  <z,  and  set  off  on  this  perpendicular  a  c  equal  to  a  B ;  draw 
through  B  c  a  line,  wrhich  will  make  with  D  E  the  required  angle  of  45°. 

PEOB.  XTT. — To  divide  a  given 
angle,  as  B  A  C  (fig.  96),  into  two 


Fig.  96. 


From  the  point  A,  or  vertex  of 
the  angle,  with  any  radius  describe 
an  arc  b  c ;  from  5  and  c,  the  inter- 
sections of  the  arc  with  the  sides  of 
the  angle,  with  any  radius  greater 


GEOMETRICAL  PROBLEMS. 


than  half  the  arc  I  c,  describe  two  arcs  intersecting  each  other,  as  at  D ; 
through  A  and  D  draw  a  line  which  will  bisect  or  divide  into  two  equal 
parts  the  angle  BAG. 

PKOB.  XIII.: — To  bisect  the  angle  contained  between  two  lines,  as  AB  and 
CD  (fig.  97),  when  the  intersecting  point  or  vertex  of  the  angle  is  not  on  the 
drawing. 

Set  off  a  point  5  at  any  convenient  distance  from  A  B,  and  through  this 
point  draw  a  parallel  to 
A  B ;  at  the  same  distance 
from  C  D  draw  a  paral- 
lel ;  extend  these  parallels 
till  they  intersect  at  c  j  bi- 
sect the  angle  5  c  d  by  c  a, 
which  will  also  bisect  the 
angle  contained  between 
the  lines  A  B  and  C  D. 

PKOB.  XIV. — Through  two  given  points,  as  B  and  C  (fig.  98),  to  describe 
an  arc  of  a  circle  with  a  given  radius. 

From  B  and  0,  with  an  opening  of  the  dividers  equal  to  the  given 
radius,  describe  two  arcs  crossing  at  A ;  from  A,  as  a  centre,  with  the  same 
radius  describe  an  arc  which  will  be  the  one  required.  It  is  to  be  observed, 
that  there  are  two  points  A,  one  above  and  one  below  the  line  B  C,  from 
which,  as  centres,  arcs  can  be  described  with  the  given  radius,  and  passing 
through  B  and  0. 


Fig.  9T. 


Fig.  93. 


Fig.  99. 


PEOB.  XY.  —  To  find  the  centre  of  a  given  circle,  or  of  an  arc  of  a  circle. 

Of  a  circle  (fig.  99).  —  Draw  the  chord  A  B,  bisect  it  by  the  perpendic- 
ular C  D,  whose  extremities  lie  in  the  circumference,  and  bisect  C  D  for 
the  centre  G  of  the  circle. 

Of  an  arc,  or  of  a  circumference  (fig.  100).  —  Select  the  points  A,  B,  C,  in 
the  circumference  well  apart  ;  with  one  radius  describe  arcs  from  these 


50 


GEOMETRIC! AT.    PKOBLEMS. 


three  points,  cutting  each  other ;  and  draw  the  two  lines  D  E,  F  G,  through 
their  intersections  :  the  point  O,  where  they  cut,  is  the  centre  of  the  circle 


or  arc. 


Fig.  100. 

PEOB.  XYI. — To  describe  a  circle  passing  through  three  given  points. 
Join  the  given  points  A,  B,  C  (fig.  100),  and  proceed  as  in  last  problem 
to  find  the  centre  O,  from  which  the  circle  may  be  described. 

This  problem  is  of  utility  :  in  striking  out  the  circular  arches  of  bridges 
upon  centering,  when  the  span  and  rise  are  given ;  describing  shallow 
pans,  or  dished  covers  of  vessels ;  or  finding  the  diameter  of  a  fly-wheel,  or 
any  other  object  of  large  diameter,  when  only  a  part  of  the  circumference 
is  accessible. 

PEOB.  XVII. — To  describe  a  circle  passing  through  three  given  points, 
when  the  centre  is  not  available. 

~Lst  Method  (fig.  101). — From  the  extreme  points  A  B,  as  centres,  de- 
scribe arcs  A  H,  B  G.  Through  the  third  point  C  draw  A  E  and  B  F, 
cutting  the  arcs.  Divide  A  F  and  B  E  into  any  number  of  equal  parts, 
and  set  off  a"  series  of  equal  parts  of  the  same  length  on  the  upper  portions 
of  the  arcs  beyond  the  points  E  F.  Draw  straight  lines  B  L,  B  M,  &c.,  to 
the  divisions  in  A  F  ;  and  A  I,  AK,  &c.,  to  the  divisions  in  E  G :  the  suc- 
cessive intersections  ~N,  O,  &c.,  of  these  lines,  are  points  in  the  circle  re- 
quired, between  the  given  points  A  and  C,  which  may  be  filled  in  accord- 
ingly. Similarly,  the  remaining  part  of  the  curve  B  C  may  be  described. 

2d  Method  (fig.  102).— 
Let  A,  D,  B,  be  the  given 
points ;  draw  AB,  AD,  DB, 
and  «/,  parallel  to  A  B.  Di- 
vide D  A  into  a  number  of 
equal  parts,  1,  2,  3,  &c., 


GEOMETRICAL    PROBLEMS. 


51 


and  from  D  describe  arcs  through  these  points  to  meet  ef.  Divide  the  arc 
A  e  into  the  same  number  of  equal  parts,  and  draw  straight  lines  from  D  to  the 
points  of  division.  The  intersections  of  these  lines  successively  with  the 
arcs  1,  2,  3,  &c.,  are  points  in  the  circle,  which  may  be  filled  in  as  before. 

Note. — The  second  method  is  not  perfectly  true,  but  sufficiently  so  for 
arcs  less  than  one-fourth  of  a  circle. 

To  describe  the  arc  mechanically  with  three  strips  of  board  forming  a 
triangle. — Insert  two  stiff  pins  or  nails  at  A  and  B ;  place  the  strips  as 
shown  in  fig.  103, 
one  against  the  pins 
at  A  and  B,  and 
having  D  at  their 
intersection;  fasten 
the  two  strongly  to- 
gether at  this  point  and  at  the  base  of  the  triangle  by  the  third  strip ;  plac- 
ing the  pencil  at  D,  and  keeping  the  edges  against  A  and  B,  moving  the 
triangle  to  the  right  and  left,  the  pencil  will  describe  the  circle. 

PROB.  XVIII. — To  draw  a  tangent  to  a  circle  from  a  given  point  in  the 
circumference. 

\st  Method. — Through  the  given  point  A  (fig.  104)  draw  the  radial  line 
A  C,  and  the  perpendicular  F  G  for  the  tangent  required. 


Fig.  103. 


Fig.  104  Fig.  105. 

%d  Method. — From  A  (fig.  105)  set  off  equal  segments,  A  B,  A  D ; 
join  B  D,  and  draw  A  E  parallel  to  it,  for  the  tangent.  This  method  is 
useful  when  the  centre  is  inaccessible. 

PROB.  XIX. — To  draw  tangents  to  a  circle  from  a  point  without  it  (fig. 
106). 

Draw  A  C  from  the  given  point  A  to  the  centre  of  the  circle,  bisect  it 
at  D,  from  D  describe  an  arc  through  C,  cutting  the  circle  at  E  and  F. 
Draw  A  E,  A  F  for  the  required  tangents. 

To  construct  within  the  sides  of  an  angle  a  circle  tangent  to  these  sides, 
at  a  given  distance  from  the  vertex. — In  fig.  107,  to  describe  a  circle  or  arc 
tangent  at  a  and  5,  equally  distant  from  the  vertex  A  ;  draw  perpendicu- 


52 


GEOMETRICAL  PROBLEMS. 


lars  to  A  C  at  a,  and  to  A  B  at  I ;  the  intersection  of  these  will  be  the 
centre  of  the  required  circle. 


Fig.  106. 


Fig.  10T. 


In  the  same  fig.,  to  find  the  centre,  the  radius  being  given,  and  not  the 
points  a  and  5. — Draw  parallels  to  A  C  and  A  B  at  a  distance  equal  to  the 
given  radius,  and  their  intersection  will  be  the  centre  required. 

PROB.  XX. — To  describe  a  circle  from  a  given  point  to  touch  a  given 
circle  (figs.  108, 109). 

D  E  being  the  given  circle,  and  B  the  point,  draw  from  B  to  the  centre  C, 
and  produce  it,  if  necessary,  to  cut  the  circle  at  A,  and  with  B  A  as  radius 
describe  the  circle  F  G,  touching  the  given  circle.  The  operation  is  the 
same  whether  the  point  B  be  within  or  without  the  circle. 


Fig.  108. 


Fig.  109. 


It  will  be  remarked  that,  in  all  cases  of  circles  tangential  to  each  other, 
their  centres  and  their  points  of  contact  must  lie  in  the  same  straight  line. 

PROB.  XXL— 
To  draw  tangents 
to  two  given  circles. 
1st  Method  (fig. 
110).— Draw  the 
straight  line  ABC 
through    the    cen- 
Fig.no.  tres    of    the    two 


GEOMETRICAL    PROBLEMS. 


53 


given  circles  ;  from  the  centres  A,  B,  draw  parallel  radii  A  D,  B  E,  in 
the  same  direction  ;  join  D  E,  and  produce  it  to  meet  the  centre  line  at  C, 
and  from  C  draw  tangents  to  one  of  the  circles  by  Prob.  XIX.  Those  tan- 
gents will  touch  both  circles,  as  required. 

2d  Method  (fig.  111). — Draw  A  B,  and  in  the  larger  circle  draw  any 
radius  A II,  on  which  set  off  II G  equal  to  the  radius  of  the  smaller  circle ; 


Fig.  111. 

on  A  describe  a  circle  with  the  radius  A  G,  and  draw  tangents  B  I,  B  K,  to 
this  circle  from  the  other  centre  B  ;  from  A  and  B  draw  perpendiculars  to 
these  tangents,  and  join  C,  D,  and  E,  F,  for  the  required  tangents. 

Note. — The  second  method  is  useful  when  the  diameters  of  the  circles 
are  nearly  equal. 

PROB.  XXII. — Between  two  inclined  lines  to  draw  a  series  of  circles 
touching  these  lines  and  touching  each  other  (fig.  112). 

Bisect  the  inclination  of  the  given  lines  A  B,  0  D,  by  the  line  K  O  ; 
this  is  the  centre  line  of  the  circles  to  be  inscribed.  From  a  point  P  in  this 
line  draw  the  perpendicular  P  B 
to  the  line  A  B,  and  from  P  de- 
scribe the  circle  B  D  touching  the 
given  lines  and  cutting  the  centre 
line  at  E  ;  from  E  draw  E  F  per- 
pendicular to  the  centre  line,  cut- 
ting A  B  at  F,  and  from  F  describe 

an  arc  E  G,  cutting  A  B  at  G ;  Fig.  n2. 

draw  G  II  parallel  to  B  P,  giving  H  the  centre  of  the  second  touching 
circle,  described  with  the  radius  H  E  or  H  G.  By  a  similar  process  the 
third  circle  I  N  is  determined.  And  so  on. 

Inversely,  the  largest  circle  may  be  described  first,  and  the  smaller 
ones  in  succession. 

Note. — This  problem  is  of  frequent  use  in  scroll  work. 

PROB.  XXIII. — Between  two  inclined  lines  to  draw  a  circular  segment 
to  jill  up  the  angle,  and  touching  the  lines  (fig.  113). 


GEOMETRICAL    PROBLEMS. 


Let  A  B,  D  E,  be  the  inclined  lines  ;  bisect  the  inclination  by  the  line 
F  C,  and  draw  the  perpendicular  A  F  D  to  define  the  limit  within  which 
the  circle  is  to  be  drawn.  Bisect  the  angles  A  and  D  by  lines  cutting  at 
C,  and  from  C  with  radius  C  F,  draw  the  arc  H  F  G  as  required. 


Fig.  114 


PROB.  XXIY. — To  -fill  up  the  angle  of  a  straight  line  and  a  circle,  with 
a  circular  arc  of  a  given  radius  (fig.  114). 

In  the  given  circle  A  D  draw  a  radius  C  B  and  produce  it,  set  off  B  E 
equal  to  the  radius  of  the  required  arc,  and  on  the  centre  C  with  the  radius 
C  E,  describe  the  arc  E  F.  Draw  G  F  parallel  to  the  given  line  H  I,  at 
the  distance  G  H  equal  to  the  radius  of  the  required  arc,  and  cutting  the 
arc  E  F  at  F.  Then  F  is  the  required  centre ;  draw  the  perpendicular  F  I, 
and  the  radius  F  C  cutting  the  circle  at  A,  and  with  the  radius  F  A  or  F I 
describe  the  arc  A I  as  required. 

PROB.  XXY. — To  fill  up  the  angle  of  a  straight  line  and  a  circle,  with 
a  circular  arc  to  join  the  circle  at  a  given  point  (fig.  115). 

In  the  given  circle  draw  the  radius  A  and  produce  it ;  at  A  draw  a 

tangent  meeting  the  given  line  at 
D ;  bisect  the  angle  A  D  E  so  formed 
with  a  line  cutting  the  radius  at  F  ; 
and  on  the  centre  F  describe  the 
arc  A  E  as  required. 

PROB.  XXVL— To  describe  a 
circular  arc  joining  two  circles,  and 
to  touch  one  of  them  at  a  given 
point  (fig.  116). 

Let  A  B  and  F  G  be  the  given 
circles,  to  be  joined  by  an  arc 

touching  one  of  them  at  F.    Draw  the  radius  E  F,  and  produce  it  both 
ways  ;  set  off  F  H  equal  to  the  radius  A  C  of  the  other  circle  ;  join  C  H 


\ 


GEOMETRICAL   PROBLEMS. 


55 


and  bisect  it  with  the  perpendicular  L  I  cutting  E  F  at  I ;  on  the  centre  I 
with  radius  I  F  describe  the  arc  F  as  required. 


Fig.  nr. 

PEOB.  XXTR—Tofindthe  arc  which  shall 
be  tangent  to  a  given  point  A  on  a  straight  line, 
and  pass  through  a  given  point  outside  the  line 
(fig.  117). 

Erect  at  A  a  perpendicular  to  the  given  line  ;  connect  C  A,  and  bisect 
it  by  a  perpendicular ;  the  intersection  of  the  two  perpendiculars  at  a  will 
be  the  centre  of  the  required  arc. 

PKOB.  XXVIII. — To  connect  two  parallel  lines  ty  a  reversed  curve  com- 
posed of  two  arcs  of  equal  radius,  and  tangent  to  the  lines  at  given  points, 
as  at  A  and  B  (fig.  118). 

Join  A  B,  and  divide  it  into  two  equal  parts  at  C  ;  bisect  C  A  and  C  B 
by  perpendiculars  ;  at  A  and  B  erect  perpendiculars  to  the  given  lines,  and 
the  intersections  a  and  b  will  be  the  centres  of  the  arcs  composing  the  re- 
quired curve. 


Fig.  118. 


Fig.  119. 


PROB.  XXIX. — To  join  two  given  points,  as  A  and  B  (fig.  119),  in  two 
given  parallel  lines  by  a  reversed  curve  of  two  equal  arcs,  whose  centres  lie 
in  the  parallels. 

Join  A  B,  and  divide  it  in  equal  parts  at  C,  as  above.  Bisect  also  A  C 
and  B  C  by  perpendiculars  ;  the  intersections  a  and  b  of  the  parallel  lines 
by  these  perpendiculars  will  be  the  centre  of  the  required  arcs. 

PROB.  XXX. — On  a  given  line,  as  A  B  (fig.  120),  to  construct  a  com- 
pound curve  of  three  arcs  of  circles,  the  radii  of  the  two  side  ones  being 


56 


GEOMETRICAL   PROBLEMS. 


equal  and  of  a  given  length,  and  their  centres  in  the  given  line  j  the  central 
arc  to  pass  through  a  given  point,  as  C,  on  the  perpendicular  bisecting  the 
given-line,  and  tangent  to  the  other  two  arcs. 

Draw  the  perpendicular  CD;  lay  off  A  a,  B  b,  and  C  c,  eacli  equal  to 

the  given  radius  of  the 
side  arcs ;  join  a  c; 
bisect  a  c  by  a  per- 
pendicular ;  the  inter- 
section of  this  line 
with  the  perpendicu- 
lar C  D  will  be  the 
required  centre  of  the 
central  arc.  Through 
a  and  b  draw  the  lines 
D  e  and  D  e' ;  from  a 
and  b  with  the  given  radius,  equal  to  a  A,  b  B,  describe  the  arcs  A  e  and 
B  e',  from  D  as  a  centre,  with  a  radius  equal  to  C  D,  and  consequently  by 
construction  D  e  and  D  e',  describe  the  arc  e  C  e',  and  we  have  the  com- 
pound curve  required. 

For  the  -construction  of  compound  curves  of  five  arcs,  see  construction 
of  ellipses,  page  72. 


0,                          X 

\    \ 

/ 

\    \ 

/ 

\  \ 

/ 

\\ 

/ 

li 

/ 

Fig.  120. 

PROBLEMS  ON  CIRCLES  AND  RECTILINEAR  FIGURES. 

PROB.  XXXI. — To  construct  a  triangle  upon  a  given  straight  line  or 
base,  the  length  of  the  two  sides  being  given. 

First,  an  equilateral  triangle  (fig.  121).  On  the  ends  A  B  of  the  given 
line,  with  A'B  as  radius,  describe  arcs  cutting  at  C,  and  draw  A  C,  B  C  ; 
then  A  B  C  is  the  triangle  required. 


Fig.  121.  Fig.  122. 

Second,  when  the  sides  are  unequal  (fig.  122).    Let  A  D  be  the  base, 
and  B  and  C  the  two  sides.     On  either  end,  as  A,  of  the  base-line,  with  the 


GEOMETRICAL    PROBLEMS. 


57 


line  B  as  radius,  describe  an  arc ;  and  on  D,  with  C  as  radius,  cut  the  arc 
at  E.     Draw  A  E,  E  D,  then  A  E  D  is  the  triangle  as  required. 

This  construction  is  used  also  to  find  the  position  of  a  point,  when  its 
distances  are  given  from  two  other  given  points,  whether  joined  by  a  line 
or  not. 

PKOB.  XXXII. — To  construct  a  square  or  a  rectangle  upon  a  given 
straight  line. 

First,  a  square  (fig.  123).  Let  A  B  be  the  given  line  ;  on  A  and  B  as 
centres,  with  the  radius  A  B,  describe  arcs 
cutting  at  C  ;  on  C,  with  the  same  radius, 
describe  arcs  cutting  the  others  at  D  and 
E ;  and  on  D  and  E,  cut  these  at  F  G. 
Draw  A  F,  B  G,  cutting  the  arcs  at  H,  I ; 
and  join  H  I  to  form  the  square  as  re- 
quired. 

Second,  a  square  or  rectangle  (fig.  124). 
To  the  base  E  F  draw  perpendiculars  E  H, 
F  G,  equal  to  the  sides,  and  join  G  H  to  complete  the  rectangle. 
•     When  the  centre  lines  of  the  square  or  rectangle  are  given,  the  figure 
may  be  described  as  follows  : — 

c. 
it  e  K          it  L 


Fig.  123. 


Fig.  124 

Let  A  B  and  C  D  (fig.  125)  be  the  centre  lines,  perpendicular  to  each 
other,  and  E  the  middle  point  of  the  figure  ;  set  off  E  F,  E  G,  equal  each 
to  the  half  length  of  the  rectangle,  and  E  H,  E  J,  each  equal  to  half  the 
height.  On  the  centres  H,  J,  with  a  radius  equal  to  the  half  length,  de- 
scribe arcs  on  both  sides  ;  and  on  F,  G,  with  a  radius  of  half  the  height, 
cut  these  arcs  at  K,  L,  M,  N.  Join  the  four  intersections  so  formed,  to 
complete  the  rectangle. 

PKOB.  XXXIII. — To  construct  a  parallelogram,  of  which  the  sides  and 
one  of  the  angles  are  given  (fig.  126). 

Let  A  and  B  be  the  lengths  of  the  two  sides,  and  C  the  angle ;  draw  a 
straight  line,  and  set  off  D  E  equal  to  A ;  from  D  draw  D  F  equal  to  B, 
and  forming  an  angle  with  D  E  equal  to  C ;  from  E  with  D  F  as  radius, 


58 


GEOMETRICAL   PROBLEMS. 


describe  an  arc,  and  from  F  with  D  E  as  radius,  cut  the  arc  at  G,  and 
draw  F  G  and  E  G,  to  complete  the  parallelogram.  Or,  the  remaining 
sides  may  be  drawn  parallel  to  D  E  and  D  F,  cutting  at  G,  and  the  figure 
is  thus  completed. 


Fig.  126. 


Fig.  127. 


PROB.  XXXIY.— To  describe  a  circle  about  a  triangle  (fig.  127). 

Bisect  two  of  the  sides  A  B,  A  C,  of  the  triangle  at  E,  F  ;  from  these 
points  draw  perpendiculars  cutting  at  K.  From  the  centre  K,  with  K  A 
as  radius,  describe  the  circle  A  B  C,  as  required. 

PROB.  XXXV. — To  inscribe  a  circle  in  a  triangle  (fig.  128). 

Bisect  two  of  the  angles  A,  C,  of  the  triangle  A  B  C,  by  lines  cutting 
at  D  ;  from  D  draw  a  perpendicular  D  E  to  any  side,  as  A  C  ;  and  with 
D  E  as  radius,  from  the  centre  D,  describe  the  circle  required. 

"When  the  triangle  is  equilateral,  the  centre  of  the  circle  is  more  easily 
found  by  bisecting  two  of  the  sides,  and  drawing  perpendiculars,  as  in  the 
previous  problem.  Or,  draw  a  perpendicular  from  one  of  the  angles  to 
the  opposite  side,  and  from  the  side  set  off  one-third  of  the  length  of  the 
perpendicular. 


E 

Fig.  128. 


Fig.  129. 


PROB.  XXXYI. — To  inscribe  a  square  in  a  circle  ;  and  to  describe  a 
circle  about  a  square  (fig.  129). 


GEOMETRICAL    PROBLEMS. 


59 


To  inscribe  the  square.  Draw  two  diameters  A  B,  C  D,  at  right  angles, 
and  join  the  points  A,  B,  C,  D,  to  form  the  square  as  required. 

To  describe  the  circle.  Draw  the  diagonals  A  B,  C  D,  of  the  given 
square,  cutting  at  E ;  on  E  as  a  centre,  with  E  A  as  radius,  describe  the 
circle  as  required. 

In  the  same  way,  a  circle  may  be  described  about  a  rectangle. 

PROB.  XXX YIL — -To  inscribe  a  circle  in  a  square  ;  and  to  describe  a 
square  about  a  circle  (fig.  130). 

To  inscribe  the  circle.  Draw  the  diagonals  A  B,  CD,  of  the  given 
square,  cutting  at  E  ;  draw  the  perpendicular  E  F  to  one  of  the  sides,  and 
with  the  radius  E  F,  on  the  centre  E,  describe  the  circle. 

To  describe  the  square.  Draw  two  diameters  A  B,  C  D,  at  right  angles, 
and  produce  them  ;  bisect  the  angle  D  E  B  at  the  centre  by  the  diameter 
F  G,  and  through  F  and  G  draw  perpendiculars  A  C,  B  D,  and  join  the 
points  A,  D,  and  B,  C,  where  they  cut  the  diagonals,  to  complete  the  square. 

PROB.  XXXYIII. — To  inscribe  a  pentagon  in  a  circle  (fig.  131). 

Draw  two  diameters  A  C,  B  D,  at  right  angles ;  bisect  A  O  at  E,  and 


Fig.  131. 


from  E  with  radius  E  B,  cut  A  C  at  F  ;  from  B,  with  radius  B  F,  cut  the 
circumference  at  G  H,  and  with  the  same  radius  step  round  the  circle  to  I 
and  K  ;  join  the  points  so  found  to  form  the  pentagon. 

PROB.  XXXIX. — To  construct  a  regular  hexagon  upon  a  given  straight 
line  (fig  132). 

From  A  and  B,  with  a  radius  equal  to  the  given  line,  describe  arcs 
cutting  at  g  /  from  ^,  with  the  radius  g  A,  describe  a  circle  ;  with  the  same 
radius  set  off  from  A  the  arcs  A  G,  G  F,  and  from  B  the  arcs  B  D,  D  E. 
Join  the  points  so  found  to  form  the  hexagon. 

PROB.  XL. — -To  inscribe  a  regular  hexagon  in  a  circle  (fig.  133). 

Draw  a  diameter  A  B,  from  A  and  B  as  centres,  with  the  radius  of  the 
circle  A  C,  cut  the  circumference  at  D,  E,  F,  G ;  draw  straight  lines  A  D, 
D  E,  &c.,  to  form  the  hexagon. 


60 


GEOMETEICAL   PROBLEMS. 


The  points  of  contact,  D,  E,  &c.,  may  also  be  found  by  setting  off  the 
radius  six  times  upon  the  circumference. 


Fig.  134. 

PEOB.  XLI. — To  describe  a  regular  hexagon  about  a  circle  (fig.  134). 

Draw  a  diameter  A  B  of  the  given  circle ;  with  the  radius  A  D  from  A 
as  a  centre,  cut  the  circumference  at  C  ;  join  A  C,  and  bisect  it  with  the 
radius  D  E ;  through  E  draw  F  G  parallel  to  A  C,  cutting  the  diameter  at 
F,  and  with  the  radius  D  F  describe  the  circle  F  II.  Within  this  circle 
describe  a  regular  hexagon  by  the  preceding  problem ;  the  figure  will  touch 
the  given  circle  as  required. 

PEOB.  XLIE. — To  construct  a  regular  octagon  upon  a  given  straight  line 
(fig.  135). 

Produce  the  given  line  A  B  both  ways,  and  draw  perpendiculars  A  E, 
B  F  ;  bisect  the  external  angles  at  A  and  B  by  the  lines  A  H,  B  C,  which 
make  equal  to  A  B  ;  draw  C  D  and  H  G  parallel  to  A  E  and  equal  to  A  B ; 
and  from  the  centres  G,  D,  with  the  radius  A  B,  cut  the  perpendiculars  at 
E,F,  and  draw  E  F  to  complete  the  octagon. 


,  E 


A 


Fig.  136. 


PBOB.  XLIII. — To  convert  a  square  into  a  regular  octagon  (fig.  136). 

Draw  the  diagonals  of  the  square  cutting  at  e'}  from  the  corners  A,  B,  C,  D, 
with  A  e  as  radius,  describe  arcs  cutting  the  sides  at  g  h,  &c. ;  join  the 
points  so  found  to  complete  the  octagon. 


GEOMETRICAL   PROBLEMS. 


Gl 


PKOB.  XLIV. — To  inscribe,  a  regular  octagon  in  a,  circle  (fig.  137"). 
Draw  two  diameters  A  0,  B  D,  at  right  angles  ;  bisect  the  arcs  A  B, 
B  C,  &c.,  at  e,fj  &c. ;  and  join  A  e,  e  B,  &c.,  for  the  inscribed  figure. 


r  — 7f 


Fig.  1ST. 


Fig.  133. 


PEOB.  XLV. — To  describe  a  regular  octagon  about  a  circle  (fig.  138). 

Describe  a  square  about  the  given  circle  A  B ;  draw  perpendiculars 
h  k,  &c.,  to  the  diagonals,  touching  the  circle.  The  octagon  so  formed  is 
the  figure  required. 

Or,  to  find  the  points  h,  k,  &c.,  cut  the  sides  from  the  corners  of  the 
square,  as  in  Prob.  XLIII. 

PKOB.  XLYI. — To  inscribe  a  circle  within  a  regular  polygon. 

When  the  polygon  has  an  even  number  of  sides,  as  in  fig.  139,  bisect 
two  opposite  sides  at  A  and  B,  draw  A  B,  and  bisect  it  at  C  by  a  diagonal 
D  E  drawn  between  opposite  angles ;  with  the  radius  C  A  describe  the 
circle  as  required. 

"When  the  number  of  sides  is  odd,  as  in  fig.  140,  bisect  two  of  the  sides 
at  A  and  B,  and  draw  lines  A  E,  B  D,  to  the  opposite  angles,  intersecting 
at  0  ;  from  C  with  C  A  as  radius,  describe  the  circle  as  required. 


PKOB.  XL VII. — To  describe  a  circle  without  a  regular  polygon. 

When  the  number  of  sides  is  even,  draw  two  diagonals  from  opposite 
angles,  like  D  E  (fig.  139),  to  intersect  at  C ;  and  from  C  with  C  D  as 
radius,  describe  the  circle  required. 


62  GEOMETRICAL,   PROBLEMS. 

When  the  number  of  sides  is  odd,  find  the  centre  C  (fig.  140)  as  in  last 
problem,  and  with  C  D  as  radius  describe  the  circle. 

The  foregoing  selection  of  problems  on  regular  figures  is  the  most  use- 
ful in  mechanical  practice  on  that  subject.  Several  other  regular  figures 
may  be  constructed  from  them  by  bisection  of  the  arcs  of  the  circumscrib- 
ing circles.  In  this  way  a  decagon,  or  ten-sided  polygon,  may  be  formed 
from  the  pentagon  by  the  bisection  of  the  arcs  in  Prob.  XXXVIIL,  fig. 
131.  Inversely,  an  equilateral  triangle  may  be  inscribed  by  joining  the 
alternate  points  of  division  found  for  a  hexagon. 

The  constructions  for  inscribing  regular  polygons  in  circles  are  suitable 
also  for  dividing  the  circumference  of  a  circle  into  a  number  of  equal 
parts.  To  supply  a  means  of  dividing  the  circumference  into  any  number 
of  parts,  including  cases  not  provided  for  in  the  foregoing  problems,  the 
annexed  table  of  angles  relating  to  polygons,  expressed  in  degrees,  will  be 
found  of  general  utility.  In  this  table,  the  angle  at  the  centre  is  found  by 


TABLE  or  POLYGONAL  ANGLES. 


Number  of  Sides  of  Regular 
Polygon  ;    or    number   of 
equal  parts  of  the  circum- 

Angle at 
Centre. 

Number  of  Sides  of  Regular 
Polygon. 

Angle  at 
Centre. 

ference. 

No. 

Degrees. 

No. 

Degrees. 

3 

120 

12 

30 

4 

90 

13 

27  rV 

5 

72 

14 

25| 

6 

60 

15 

24 

7 

51.3 

16 

224- 

8 

45  7 

17 

21A 

9 

40 

18 

20 

10 

36 

19 

18H 

11 

32T«T 

20 

18 

dividing  360°,  the  number  of  degrees  in  a  circle,  by  the  number  of  sides 
in  the  polygon ;  and  by  setting  off  round  the  centre  of  the  circle  a  suc- 
cession of  angles  by  means  of  the  protractor,  equal  to  the  angle  in  the 
table  due  to  a  given  number  of  sides :  the  radii  so  drawn  will  divide  the 
circumference  into  the  same  number  of  parts.  The  triangles  thus  formed 
are  termed  the  elementary  triangles  of  the  polygon. 

PROB.  XLYIII. — To  inscribe  any  regular  polygon  in  a  given  circle  ;  or  to 
divide  the  circumference  into  a  given  number  of  equal  parts,  by  means  of 
the  angle  at  the  centre  (fig.  141). 


GEOMETRICAL    PROBLEMS. 


63 


Suppose  the  circle  is  to  contain  a  hexagon,  or  is  to  be  divided  at  the 
circumference  into  six  equal  parts.  Find  the  angle 
at  the  centre  for  a  hexagon,  or  60°  ;  draw  any  ra- 
dius B  C,  and  set  off  by  a  protractor  or  otherwise 
the  angle  at  the  centre  C  B  D,  equal  to  60°  ;  then 
the  interval  C  D  is  one  side  of  the  figure,  or  seg- 
ment of  the  circumference ;  and  the  remaining 
points  of  division  may  be  found  either  by  stepping 
along  the  circumference  with  the  distance  C  D  in 
the  dividers,  or  by  setting  off  the  remaining  five 
angles  of  60°  each  round  the  centre. 


Fig.  141. 


THE   USE   OF   TIIE    T    SQUARE   AND   TRIANGLE   IN   THE   CONSTRUCTION   OF   SOME 
OF   THE   FOREGOING    PROBLEMS. 

From  the  description  of  the  T  square  it  may  be  seen,  that  by  sliding 
the  stock  along  two  contiguous  edges  of  the  board,  the  left  hand  and 
bottom  edges,  any  number  of  parallel  and  perpendicular  lines  may  be 
drawn.  In  so  far,  therefore,  the  T  square  supersedes  the  application  of  all 
the  problems  for  drawing  parallels  and  perpendiculars,  coinciding  in 
direction  with  the  edges  of  the  board ;  for  the  square  need  only  be  set 
with  its  edge  coincident  with  the  points  through  which  the  line  is  to  be 
drawn,  and  the  pen  or  pencil  drawn  along  the  edge  will  describe  the  line 
required.  When  the  perpendiculars  or  upright  lines  are  of  short  length, 
the  triangle  and  ruler  are  used.  For  this  purpose,  the  triangle  or  set 
square  of  60°  is  preferable  to  that  of  45°,  as  it  is  longer  and  lighter. 

When  the  lines  to  be  drawn  do  not  coincide  in  direction  with  the  edges 
of  the  board,  the  square  may  be  adjusted  with  its  bevel  stock  to  the 
obliquity  required,  and  the  lines  may  be  drawn  as  before.  This  is  proba- 
bly the  best  plan  when  the  oblique  lines  are  numerous  or  extensive.  In 
most  cases,  however,  oblique  lines  are  only  occasional,  and  when  their 
position  is  given,  they  may  be  drawn  with  a  straight-edge.  When  the 
oblique  parallels  and  perpendiculars  are  short,  as  in  oblique  framing,  short 
rods  or  bars,  bolt-heads,  and  the  like,  the  com- 
bined use  of  the  straight-edge  and  triangle  is 
expedient.  Square  figures  may  be  described 
on  a  given  centre,  at  one  setting  of  the  / 
straight-edge,  as  in  the  drawing  of  the  head  Fig.  142. 

of  a  square  nut  n  (fig.  ,142).     From  the  given  centre,  with  a  radius  equal 


64 


GEOMETRICAL.  PROBLEMS. 


to  half  the  side  of  the  square,  describe  a  circle,  and  with  the  aid  of  the  tri- 
angle draw  lines  tangent  to  four  sides. 

To  draw  an  octagon,  apply  the  set-square  of  45°  to  the  corners,  after 
completing  a  square  figure,  and  draw  tangents  to  the  inscribed  circle,  as, 
for  example,  the  line  h  k  (fig.  138). 

To  draw  an  equilateral  triangle  upon  a  given  line  A  B  (fig.  143),  it  is 
only  necessary  to  apply  the  slant  edge  of  the  set-square  of  60°  to  each  end 


Fig.  143. 


Fig.  144. 


Fig.  145. 


of  the  base,  with  the  short  side  b  c  applied  to  the  square-blade,  and  to 
draw  the  two  sides  A  0,  B  C.  If  the  given  side  A  B  be  upright  (fig.  144), 
apply  the  long  side  a  l>  to  the  straight-edge,  and  draw  as  before. 

To  draw  a  regular  hexagon  about  a  circle,  with  two  of  its  sides  parallel 
to  the  lower  edge  of  the  board  :  draw  the  centre  line  A  B  (fig.  145),  and 
the  upper  and  lower  sides  D  E,  F  G,  touching  the  circle,  and  apply  the 
triangle  of  60°  touching  the  circle  for  the  four  remaining  sides,  as  shown 
in  the  figure. 

When  the  hexagons  are  to  be  inscribed  in  the  circle,  first  draw  the 
centre  line  A  B  (fig.  133)  as  a  diameter,  and  from  the  ends  A,  B,  with  the 
set-square,  draw  four  sides  cutting  the  circle 
at  D,  E,  F,  G,  and  join  D  E,  F  G. 

The  triangles  of  45°  and  60°  are  useful  in 
setting  out  the  centre-lines  of  wheels  with  3, 
4,  6,  8,  &c.,  arms,  by  drawing  lines  through 
the  centre  of  the  wheel.  To  set  out  12  spokes 
in  a  wheel  (fig.  146)  : — Draw  two  diameters, 
A  B,  CD,  parallel  to  the  two  edges  of  the 
board  ;  in  the  quadrant  A  C,  draw  radii  E  a, 
E  J,  with  the  long  and  the  short  sides  of  the 
triangle  against  the  square-blade.  These  will 
divide  the  quadrant  equally  ;  and  the  same 
construction  being  employed  for  the  other  quarters  of  the  circle,  12  centre 


Fig.  146. 


GEOMETRICAL   PROBLEMS. 


65 


lines,  equally  distant,  will  be  described.  Should  the  triangle  be  large 
enough  to  embrace  the  whole  circle  at  once,  the  opposite  quadrants  A  C 
and  B  D  may  be  divided  with  the  same  setting  of  the  triangle. 

A  short  method  of  dividing  a  line  or  surface  into  a  number  of  equal 
parts  is  illustrated  by  fig.  147 ;  and  it  is  convenient  where  an  ordinary 
rule  does  not  evenly 
measure  the  dimen- 
sion. Suppose  the 
width  A  C  is  to  be  di- 
vided into  seven  equal 
parts,  and  that  it  mea- 
sures 7|  inches  ;  an  or- 
dinary inch  rule,  it  is 
plain,  does  not  afford  the  subdivisions  when  applied  directly ;  but  if  14 
inches  of  length,  or  double  the  number  of  parts,  be  applied  obliquely  across 
the  space  between  the  parallels  A  B,  C  D,  so  as  to  measure  it  exactly, 
then  point  off  two-inch  intervals  on  the  edge  of  the  rule,  and  in  this  way 
7  equal  subdivisions  will  be  effected,  through  which  parallels  may  be 
drawn. 


SIMPLE   APPLICATIONS   OF  REGULAR  FIGURES. 


PROB.  XLIX. — To  cover  a  surface  with  equilateral  triangles,  hexagons, 
or  lozenges. 

Describe  an  equilateral  triangle 
ABC,  and  produce  the  sides  in- 
definitely. Set  off  from  one  angle 
A,  equal  intervals  at  a,  5,  a',  V, 
&c.,  as  required;  and  through 
these  points  draw  parallels  to  each 
of  the  sides  of  the  triangle.  The 
area  will  be  covered  with  triangles 
as  required. 

For  hexagons,  or   equilateral  ^ <L b       c      d       e      B      f 

triangles   and    hexagons    on  the  Fig.ua 

same  surface,  or  lozenges,  group  the  triangles. 

PROB.  L. — To  cover  a  surface  with  octagons  and  squares. 

Draw  two  straight  lines  A  B,  A  C,  at  right  angles  ;  set  off  equal  inter- 
5 


66 


GEOMETRICAL   PROBLEMS. 


vals  Ad,de,  &c.,  on  each  line,  equal  to  the  breadth  of  the  octagon  to  be 
described,  and  through  these  points  draw  parallels  to  the  given  lines,  to 


Fig.  149. 

form  squares.    "Within  these  squares  construct  octagons,  by  Probs.  XLTTT. 
or  XLIY.,  and  finish  as  in  the  figure. 


PROBLEMS   ON   PROPORTIONAL   LINES   AND  EQUIVALENT  FIGURES. 

PROB.  LI. — To  divide  a  given  straight  line  into  two  parts  proportional 
to  two  given  lines. 

Let  A  B  (fig.  150)  be  the  line  to  be  divided ;  draw  the  straight  line 
A  D  at  any  angle  with  A  B,  and  set  off  A  E,  E  D,  equal  to  the  other  two 
given  lines.  Join  D  B,  and  draw  C  E  parallel  to  it ;  this  line  divides  A  B 
at  C  in  the  required  ratio. 

PROB.  LII. — To  divide  a  straight  line  into  any  number  of  parts  of  given 
proportions  ;  or  similarly  to  a  given  straight  line. 

Let  A  B  (fig.  151)  be  the  line  to  be  divided.    Draw  B  G  at  any  angle 


AC  JJ 

Fig.  150. 


H    I  A 

Fig.  151. 


with  it,  and  set  off  by  any  convenient  scale,  B  C,  C  D,  &c.,  to  G,  respec- 
tively, equal  to  the  given  divisions.    Join  A  G,  and  from  the  points  of 


GEOMETRICAL   PROBLEMS. 


67 


division  of  B  G  draw  parallels  to  A  G,  cutting  it  at  H,  I,  &c.  The  paral- 
lels so  drawn  will  divide  A  B  as  required. 

PROB.  LIU. — To  find  a  fourth  proportional  to  three  given  lines. 

Draw  two  lines  I  K,  I  N  (fig.  152),  at  any  angle,  and  set  off  I  M,  I  N, 
equal  to  the  two  first  of  the  given  lines,  and  set  off  I  L  equal  to  the  third. 
Join  L  M,  and  draw  N  K  parallel  to  it.  Then  I  K  is  a  fourth  proportional 
as  required. 

The  two  first  lines  may  "be  set  off  successively  on  the  same  line,  as  from 
I  to  M,  and  from  M  to  N,  and  the  third  from  I  to  L ;  then  L  K  will  be 
the  fourth  line  required. 

PROB.  LIY. — Tojmd  a  mean  proportional  between  two  given  lines  (fig. 
153). 

Let  a  I  and  I  c  be  the  given  lines.  Set  off,  on  a  straight  line,  A  B,  B  C, 
equal  to  the  given  lines ;  bisect  A  C  at  D,  and  with  D  A  as  a  radius  describe 
the  semicircle  AE  C ;  draw  B  E  perpendicular  to  A  C,  meeting  the  circle 
at  E.  Then  B  E  is  the  mean  proportional  required. 


^,_^_,. 

/ 

E,  , 
\ 

\ 
1 

C                        D         I 

Fig.  153. 

•) 

.4 

T) 

Fig.  154 


PROB.  LY. — To  construct  a  triangle  equal  in  area  to  a  gvven 

Bisect  the  base  B  C  (fig.  154)  of  the  rectangle  at  D,  and  draw  the  per- 
pendicular D  A  equal  to  twice  the  height,  D  E,  of  the  rectangle.  Draw 
B  A,  A  C  ;  the  triangle  A  B  C  is  equal  in  area  to  the  rectangle  B  G. 

PROB.  LYI. — To  construct  a  square  equal  to  a  gwen  rectangle  (fig.  155). 

Let  A  B  0  D  be  the  rectangle;  ^  Jf  G 

produce  A  B,  and  set  off  B  E  equal 
to  the  side  B  C  of  the  rectangle ;  bi- 
sect A  E  at  K,  and  describe  a  semi- 
circle on  A  E  ;  draw  the  perpendicu- 
lar B  H,  cutting  the  circle  at  H,  and 
on  B  H  describe  the  square  B  G 
required.  Fig.  155. 

PROB.  LYII. — To  construct  a  triangle  equivalent  to  any  regular  polygon. 

Find  the  radius  of  the  circle  inscribed  in  the  polygon.    Set  off  on  a 


GEOMETRICAL   PKOBLEMS. 


right  line  a  distance  equal  to  half  the  sum  of  the  sides  of  the  polygon. 
This  distance  will  be  the  base  of  the  equivalent  triangle,  and  the  radius  of 
the  inscribed  circle  its  perpendicular  or  altitude. 


PKOBLEMS  ON  THE  ELLIPSE,   THE  PAKABOLA,   THE   HYPEBBOLA,   THE   CYCLOID, 
AND  THE  EPICYCLOID. 


PEOB.  LVlii- — To  describe  an  ellipse,  the  length  and  breadth,  or  the  two 


axes 


1st  Method  (fig.  156). — Bisect  the  transverse  axis  A  B  at  C,  and  through 

,/     V 


C  draw  the  perpendicular  D  E,  making  0  D  and  C  E  each  equal  to  half 
the  conjugate  diameter.  On  D  as  a  centre,  with  C  A  as  radius,  describe 
arcs  cutting  at  F,  F',  for  the  foci.  Divide  A  C  into  a  number  of  parts  at 
the  points  1,  2,  3,  &c.  "With  radius  A  1  on  F  and  F'  as  centres,  describe 
arcs ;  and  with  radius  B  1,  on  the  same  centres,  describe  arcs  inter- 
secting the  others  as  shown.  Repeat  the  operation  for  the  other  divisions 
of  the  transverse  axis.  The  series  of  intersections  thus  found  will  be 
points  in  the  curve,  and  they  may  be  as  numerously  found  as  is  desir- 
able ;  after  which  a  curve  traced  through  them  will  form  the  complete 
ellipse. 

2d  Method  (fig.  15T).— The  two  axes,  A  B,  D  E,  being  given.  On  A  B 
and  D  E  as  diameters  from  the  same  centre  C,  describe  circles  F  G,  H  I ; 
take  a  convenient  number  of  points,  a,  5,  &c.,  in  the  semi-circumference 
A  F  B,  and  draw  radii  cutting  the  innner  circle  at  a!  o',  &c. ;  from  «,  5, 
&c.,  draw  perpendiculars  to  A  B,  and  from  a',  V,  &c.,  draw  parallels  to 


GEOMETRICAL  PROBLEMS. 


A  B,  cutting  the  respective  perpendiculars  at  n,  o,  &c.    The  points  of  in- 
tersection so  found  are  points  in  the  curve. 

3d    Method   (fig. 
158).  —  Along    the 
straight  edge  of  a  slip 
of  stiff  paper  mark  off 
a  distance  a  c  equal 
to  A  C,  half  the  trans- 
verse axis ;  and  from 
the  same  point,  a  dis- 
tance  a  l>  equal'  to 
C  D,  half  the  conju- 
gate axis.     Place  the 
slip  so  as  to  bring  the 
point  5  on   the   line 
A  B  of  the  transverse 
axis,  and  the  point  c 
on  the  line  D  E  ;  and 
set  off  on  the  drawing 
the  position  of  the  point 
a.     Always  keeping  the 
point  T)  on  the  transverse 
axis,  and  the  point  c  on 
the  conjugate  axis,  any 
required  number  of  points 
may  be  found. 

4th  Method  (fig.  159). 
--By  the  above  method 
large  curves  may  be  de- 
scribed     continuously     by 
means  of  a  bar  m,  k,  with 
steel  points  m,  Z,  Jc,  riveted 
into  brass  slides,  adjusted  to 
the  length  of  the  semi-axes, 
and    fixed   with  set-screws. 
A  rectangular  cross  E  G, 
with  guiding  slots,  is  placed 
to  coincide  with    the    two 
axes  of  the  ellipse  A  C  and  B  H ;  by  sliding  the  points  &,  Z,  in  the  slots, 


70 


GEOMETRICAL   PROBLEMS. 


and  carrying  round  the  point  m,  the  curve  may  be  completely  described. 

If  desirable,  of  course,  a  pen  or  pencil  may  be  fixed  at  in. 

5th  Method  (fig.  160). — Given  the  two  axes  A  B,  C  D  ;  on  the  centre 

C,  with  A  E  as  radius,  describe 
an  arc  cutting  A  B  at  F  and  G, 
the  foci ;  fix  a  couple  of  pins 
into  the  transverse  axis  at  F 
and  G,  and  loop  on  a  thread 
or  cord  upon  them,  equal  in 
length,  when  looped  on,  to  A 
B,  so  as,  when  stretched,  as  per 
dot-line  FOG,  just  to  reach 
the  extremity,  C,  of  the  conju- 
gate axis.  Place  a  pencil  or 
draw-point  inside  the  cord,  as 
at  H,  and  guiding  the  pencil 

in  this  way,  keeping  the  cord  equally  on  tension,  pass   round  the  two 

points  F,  G,  and  describe  the  curve  as  required. 

This  method  is  employed  in  setting  off  elliptical  garden-plots,  walks, 

&c. 

PEOB.  LTX. — To  draw  a  tangent  to  an  ellipse  through  a  given  point 

in  the  curve  (fig.  161). 

From  the  given  point  T  draw  straight  lines  to  the  foci  F,  F' ;  produce 


F  T  beyond  the  curve  to  <?,  and  bisect  the  exterior  angle  c  T  F'  by  the  line 
T  d.  This  line  T  d  is  the  tangent  required. 

PROB.  LX. — To  draw  a  tangent  to  an  ellipse  from  a  given  point  with- 
out the  curve  (fig.  162). 

From  the  given  point  T  as  centre,  with  a  radius  equal  to  its  distance 
from  the  nearest  focus  F,  describe  an  arc ;  from  the  other  focus  F',  with 
the  transverse  axis  as  radius,  cut  the  arc  at  K,  L,  and  draw  K  F',  L  F', 


GEOMETRICAL   PROBLEMS. 


71 


cutting  the  curve  at  M,  1ST ;  then  the  lines  T  M,  T  K,  are  tangents  to  the 
curve. 


PROB.  LXL — To  describe  an 


,  by  means  of  circular 


arcs. 


First,  with  arcs  of  two  radii  (fig.  163).  Take  the  difference  of  the  trans- 
verse and  conjugate  axes,  and  set  it  off  from  the  centre  O  to  a  and  c,  on 
O  A  and  O  C  ;  draw  a  c,  and  set  off  half  a  c  to  d;  draw  d  i  parallel  to  a  c, 


set  off  O  e  equal  to  O  d,  join  e  i,  and  draw  em,  dm,  parallels  to  d  i,  i  e.  On 
centre  ra,  with  radius  m  C,  describe  an  arc  through  0,  and  from  centre  i 
describe  an  arc  through  D ;  on  centre  d,  e,  also,  describe  arcs  through  A 
and  B.  The  four  arcs  thus  described  form  approximately  an  ellipse.  This 
method  does  not  apply  satisfactorily  when  the  conjugate  axis  is  less  than 
two-thirds  of  the  transverse  axis. 

Second,  with  arcs  of  three  radii  (fig.  164).  On  the  transverse  axis  A  B, 
draw  the  rectangle  B  G,  equal  in  height  to  O  C,  half  the  conjugate  -axis. 
Draw  G  D  perpendicular  to  A  C  ;  set  off  O  K  equal  to  O  C,  and  on  A  K 
as  a  diameter,  describe  the  semicircle  A  N  K  ;  draw  a  radius  parallel  to 
O  C,  intersecting  the  semicircle  at  1ST  and  the  line  G  E  at  P  ;  extend  O  C 


GEOMETRICAL   PKOBLEMS. 


to  L  and  to  D ;  set  off  O  M  equal  to  P  IS",  and  on  D  as  a  centre,  with  a 
radius  D  M,  describe  an  arc  ;  from  A  and  B  as  centres,  with  a  radius  O  L, 


Fig.  164 

intersect  this  arc  at  a  and  I.  The  points  H,  #,  D,  5,  IF,  are  the  centres  of 
the  arcs  required  ;  produce  the  lines  a  H,  D  a,  D  5,  5  H',  and  the  spaces 
enclosed  determine  the  lengths  of  each  arc. 

This  process  works  well  for  nearly  all  proportions  of  ellipses.     It  is 
employed  in  striking  out  vaults  and  stone  bridges. 


The  Parabola. 

The  parabola  may  be  denned  as  an  ellipse  whose  transverse  axis  is  in- 
finite ;  its  characteristic  is  that  every  point  in  the  curve  is  equally  distant 
from  the  directrix  E  N  and  the  focus  F  (fig.  165). 

PBOB.  LXII. — To  construct  a  parabola  when  the  focus  and  directrix  are 
given. 

1«£  Method  (fig.  165). — Through  the  focus  F  draw  the  axis  A  B  perpen- 
dicular to  the  directrix  E  N",  and  bisect  A  F  at  e,  then  e  is  the  vertex  of  the 
curve.  Through  a  series  of  points  C,  D,  E,  on  the  directrix,  draw  parallels 
to  A  B ;  connect  these  points  C,  D,  E,  with  the  focus  F,  and  bisect  by 
perpendiculars  the  lines  F  C,  F  D,  F  E.  The  intersections  of  these  per- 
pendiculars with  the  parallels  will  give  points  in  the  curve  C'  D'  E',  through 
which  trace  the  parabola. 

2d  Method  (fig.  166).— Place  a  straight  edge  to  the  directrix  E  N,  and 


GEOMETRICAL   PROBLEMS. 


73 


apply  to  it  a  square  LEG;  fasten  at  G  one  end  of  a  cord,  equal  in  length 
to  E  G ;  fix  the  other  end  to  the  focus  F  ;  slide  the  square  steadily  along 


Fig.  165. 


Fig.  166. 


the  straight  edge,  holding  the  cord  taut  against  the  edge  of  the  square  by 
a  pencil  D,  and  it  will  describe  the  curve. 

PEOB.  LXIH. — To  construct  a  parabola,  when  the  vertex  A,  the  axis  A  B, 
and  a  point  M  of  the  curve  are  given  (fig.  167). 

Construct  the  rectangle  A  B  M  C ;  -divide  M  C  into  any  number  ol 
equal  parts,  four  for  instance ;  di- 
vide A  C  in  like  manner ;  con- 
nect A  1,  A  2,  A  3 ;  through  1' 
2'  3',  draw  parallels  to  the  axis. 
The  intersections  I,  II,  III,  of 
these  lines  are  points  in  the  re- 
quired curve. 

PKOB.  LXIY. — To  draw  a  tangent  to  a  given  point  II  of  the  parabola 
(fig.  167). 

From  the  given  point  II  let  fall  a  perpendicular  on  the  axis  at  ~b  /  ex- 
tend the  axis  to  the  left  of  A ;  make  A  a  equal  to  A  ~b  ;  draw  a  II,  and 
it  is  the  tangent  required. 

The  lines  perpendicular  to  the  tangent  are  called  normals.  To  find 
tJie  normal  to  any  point  I,  having  the  tangent  to  any  other  point  II. — Draw 
the  normal  II  c  /  from  I  let  fall  a  perpendicular  I  d  on  the  axis  A  B  ;  lay 
off  d  e  equal  to  5  c  /  connect  I  e,  and  we  have  the  normal  required.  The 
tangent  may  be  drawn  at  I  by  a  perpendicular  to  the  normal  I  e. 


Fig.  167. 


74:  GEOilETEICAI,  PEOBLEMS. 


The  Hyperbola. 

An  hyperbola  is  a  curve  from  any  point  P  in  -which,  if  two  straight 
lines  be  drawn  to  two  fixed  points,  F  F'  the  foci,  their  difference  shall 
always  be  the  same. 

PBOB.  LXY.— To  describe  an  hyperbola  (fig.  168). 

From  one  of  the  foci  F,  with  an  assumed  radius,  describe  an  arc,  and 
from  the  other  focus  F',  with  another  radius  exceeding  the  former  by  the 
given  difference,  describe  two  small  arcs,  cutting  the  first  as  at  P  and^>. 
Let  this  operation  be  repeated  with  two  new  radii,  taking  care  that  the 
second  shall  exceed  the  first  by  the  same  difference  as  before,  and  two  new 
points  will  be  determined ;  and  this  determination  of  points  in  the  curve 
may  thus  be  continued  till  its  track  is  obvious.  By  making  use  of  the 
same  radii,  but  transposing,  that  is,  describing  with  the  greater  about  F, 
and  the  less  about  F',  we  have  another  series  of  points  equally  belonging 
to  the  hyperbola,  and  answering  the  definition  ;  so  that  the  hyperbola  con- 
sists of  two  separate  branches. 


Fig.  168.  Fig.  169. 

The  curve  may  be  described  mechanically  (fig.  169). — By  fixing  a  ruler 
to  one  focus  F',  so  that  it  may  be  turned  round  on  this  point,  connect  the 
extremity  of  the  ruler  R  to  the  other  focus  F  by  a  cord  shorter  than  the 
whole  length  F  R  of  the  ruler  by  the  given  difference  ;  then  a  pencil  P 
keeping  this  cord  always  stretched,  and  at  the  same  time  pressing  against 
the  edge  of  the  ruler,  will,  as  the  ruler  revolves  around  F',  describe  an 
hyperbola,  of  which  F  F'  are  the  foci,  and  the  differences  of  distances  from 
these  points  to  every  point  in  the  curve  will  be  the  same. 

PBOB.  LXVI. — To  draw  a  tangent  to  any  point  P  of  'an  hyperbola  (fig. 
170). 


GEOMETRICAL  PROBLEMS. 


75 


,  and  from  P  let  fall  a 


Fig.  no. 


On  F'  P  lay  off  P  p  equal  to  F  P  ;  conn 
perpendicular  on  this  line  F  j?,  and 
it  will  be  the  tangent  required. 

The  three  curves,  the  ellipse, 
the  parabola,  and  the  hyperbola, 
are  called  conic  sections,  as  they 
are  formed  by  the  intersections  of  a 
plane  with  the  surface  of  a  cone 
(plate  Y). 

If  the  cone  be  cut  through  both 
its  sides  by  a  plane  not  parallel  to 
the  base,  the  section  is  an  ellipse  ; 
if  the  intersecting  plane  be  parallel  to  the  side  of  the  cone,  the  section  is  a 
parabola ;  if  the  plane  have  such  a  position,  that  when  produced  it  meets 
the  opposite  cone,  the  section  is  a  hyperbola.  The  opposite  cone  is  a 
reversed  cone  formed  on  the  apex  of  the  other  by  the  continuation  of  its 
sides. 

The  Cycloid. 

The  cycloid  is  the  curve  described  by  a  point  in  the  circumference  of  a 
circle  rolling  on  a  straight  line. 

PEOB.  LXYIL— To  describe  a  cycloid  (fig.  171). 

Draw  the  straight  line  A  B  as  the  base  ;  describe  the  generating  circle 
tangent  to  the  centre  of  this  line,  and  through  the  centre  0  draw  the  line 
E  E  parallel  to  the  base  ;  let  fall  a  perpendicular  from  0  upon  the  base  ; 


divide  the  semicircumference  into  any  number  of  equal  parts,  for  instance 
six ;  lay  off  on  A  B  and  C  E  distances  0 1',  V  2' . . .,  equal  to  the  divisions 
of  the  circumference ;  draw  the  chords  D 1,  D  2  . . .,  from  the  points  1',  2',  3' ... 
on  the  line  C  E,  with  radii  equal  to  the  generating  circle,  describe  arcs ; 


76 


GEOMETRICAL    PROBLEMS. 


from  the  points  1',  2',  3',  4',  5'  on  the  line  B  A,  and  with  radii  equal  suc- 
cessively to  the  chords  D  1,  D  2,  D  3,  D  4,  D  5,  describe  arcs  cutting  the 
preceding,  and  the  intersections  will  be  points  of  the  curve  required. 

.  2d  Method  (fig.  172).— Let 

0  9'  be  the  base  line,  0  4  9  the 
half  of  the  generating  circle  ; 
divide  the  half  circle  into  any 
number  of  equal  parts,  say  9, 
and  draw  the  chord  01,  02, 
0  3,  &e. ;  lay  off  on  the  base 
0  1',  V  2',  2'  3' . . . . ,  equal  re- 
spectively to  the  length  of  one 
of  the  divisions  of  the  half 
circle  0  1 ;  draw  through  the 

points  1',  2',  3' lines  par- 

•a£\     allel  to  the  chords  0  1,  0  2, 
Wl    03....;  the  intersections  I, 

II,  in ....  of  these  lines  are  centres  of  the  arcs  0  a,  a  5,  b  c  . . . . ,  of  which 
the  cycloid  is  composed. 


The  Epicycloid. 

The  epicycloid  is  formed  by  a  point  in  the  circumference  of  a  circle 
revolving  either  externally  or  internally  on  the  circumference  of  another 
circle  as  a  base. 

PKOB.  LXVIIL — To  describe  an  epicycloid. 

Let  us  in  the  first  place  take  the  exterior  curve.  Divide  the  circum- 
ference A  B  D  (fig.  173)  into  a  series  of  equal  parts  1,  2,  3  . . . .,  beginning 
from  the  point  A  ;  set  off  in  the  same  manner,  upon  the  circle  A  M,  A  N, 

the  divisions  1',  2',  3' equal  to  the  divisions  of  the  circumference  A  B  D. 

Then,  as  the  circle  A  B  D  rolls  upon  the  circle  A  M  A  N,  the  points  1, 2,  3 
will  coincide  successively  with  the  points  1',  2',  3' ;  and,  drawing  radii 
from  the  point  O  through  the  points  1',  2',  3',  and  also  describing  arcs  of 
circles  from  the  centre  O,  through  the  points  1,  2,  3,  . . .  .,  they  will  inter- 
sect each  other  successively  at  the  points  c,  d,  e Take  now  the  dis- 
tance 1  to  c,  and  set  it  off  on  the  same  arc  from  the  point  of  intersection,  *, 
of  the  radius  A  C ;  in  like  manner,  set  off  the  distance  2  to  d,  from  I  to  A1, 
and  the  distance  3  to  e  to  As,  and  so'  on.  Then  the  points  A1,  Aa,  A1, 
will  be  so  many  points  in  the  epicycloid  ;  and  their  frequency  may  be  in- 


GEOMETRICAL   PROBLEMS. 


77 


creased  at  pleasure  by  shortening  the  divisions  of  the  circular  arcs.  Thus 
the  form  of  the  curve  may  be  determined  to  any  amount  of  accuracy,  and 
completed  by  tracing  a  line  through  the  points  found. 

As  the  distances  1  to  c, which  are  near  the  commencement  of  the 

curve,  must  be  very  short,  it  may,  in  some  instances,  be  more  convenient 
to  set  off  the  whole  distance  i  to  1  from  <?,  and  in  the  same  way  the  distance 
b  to  2  from  d  to  Aa,  and  so  on.  In  this  manner  the  form  of  the  curve  is 
the  more  likely  to  be  accurately  defined. 


2d  Method. — To  find  the  points  in  the  curve,  find  the  positions  of  the 
centre  of  the  rolling  circle  corresponding  to  the  points  of  contact  1',  2',  3', 
&c.,  which  may  be  readily  done  by  producing  the  radii  from  the  centre  O, 

through  the  points  I/,  2',  3', to  cut  the  circle  B  C.  From  these  centres 

describe  arcs  of  a  circle  with  the  radius  of  C  A,  cutting  the  corresponding 
arcs  described  from  the  centre  0,  and  passing  through  the  points  A1,  A2, 
A3, ....  as  before. 

When  the  moving  circle  A  B  D  is  made  to  roll  on  the  interior  of  the 
circumference  A  M,  A  1ST,  as  shown  (fig.  174),  the  curve  described  by  the 
point  A  is  called  an  interior  epicycloid.  It  may  be  constructed  in  the 


78 


GEOMETRICAL  PROBLEMS. 


same  way  as  in  the  preceding  case,  as  may  be  easily  understood,  the  same 
figures  and  letters  of  reference  being  used  in  both  figures. 


Fig.  174 


The  Involute. 

The  involute  is  a  curve  traced  by  the  extremity  of  a  flexible  line  un- 
winding from  the  circumference  of  a  circle. 


PROB.  LXIX. — To  describe  an  involute. 

Divide  the  circumference  of  the  given  circle  (fig.  175)  into  any  number 


GEOMETRICAL   PROBLEMS. 


79 


of  equal  parts,  as  0, 1,  2,  3,  4, ;  at  each  of  these  points  draw  tangents  to 

the  given  circle ;  on  the  first  of  these  lay  off  the  distance  1  I/,  equal  to  the 
arc  0  1 ;  on  the  second  lay  off  2  2',  equal  to  twice  the  arc  0  1  or  the  arc 

0  2  :  establish  in  a  similar  way  the  points  3',  4',  5', as  far  as  may  be 

requisite,  which  are  points  in  the  curve  required. 

It  may  be  remarked,  that  in  all  the  problems  in  which  curves  have 
been  determined  by  the  position  of  points,  that  the  more  numerous  the 
points  thus  fixed,  the  more  accurately  can  the  curve  be  drawn. 

The  involute  curve  may  be  described  mechanically  in  several  ways. 
Thus,  let  A  (fig.  176)  be  the  cen- 
tre  of  a  wheel  for  which  the  form 
of  involute  teeth  is  to  be  found. 
Let  m  n  a  be  a  thread  lapped 
round  its  circumference,  having 
a  loop-hole  at  its  extremity  a; 
in  this  fix  a  pin,  with  which  de- 
scribe the  curve  or  involute  a  5 

A,  by  unwinding  the  thread 

gradually  from  the  circumfer-  rig.  ITS. 

ence,  and  this  curve  will  be  the  proper  form  for  the  teeth  of  a  wheel  of  the 

given  diameter. 


The  Spiral. 

The  spiral  is  the  involute  of  a  circle  produced  beyond  a  single  revolution. 

PKOB.  LXX. — To  describe  a  spiral  (figs.  5  and  6,  plate  XII). 

Divide  the  circumference  of  the  primary  into  any  number  of  equal 
parts,  say  not  less  than  eight.  To  these  points  of  division  e,f,  i,  &c.,  draw 
tangents,  and  from  these  points  draw  a  succession  of  circular  arcs ;  thus, 
from  e  as  a  centre,  with  the  radius  e  g,  equal  to  the  arc  a  e  reduced  to  a 
straight  linej  describe  the  arc  a  g  Kj  from  /",  with  the  radius  &/",  describe 
the  arc  g  h  •  from  i  the  next  arc,  and  so  on.  Continue  the  use  of  the  centres 
successively  and  repeatedly  to  the  extent  of  the  revolutions  -required. 
Thus  the  point  a  in  the  fig.  is  used  as  a  centre  for  three  arcs,  bl,em,dn. 


80 


GEOilETEICAL   PROJECTION. 


GEOMETKICAL  PKOJECTIOK 


AKCHTTECTUKAL  and  mechanical  drawing  is  generally  the  delineation  of 
bodies  by  geometrical  or  orthographic  projection ;  the  representation, on  a 
sheet  of  paper  which  has  only  two  dimensions  length  and  breadth,  of  solids 
which  have  three,  length,  breadth,  and  thickness. 

Since  the  surfaces  of  all  bodies  may  be  considered  as  composed  of 
points,  the  first  step  is  to  represent  the  position  in  space  of  a  point,  by  re- 
ferring it  planes  whose  position  is  established.  The  projection  of  a  point 
upon  a  plane  is  the  foot  of  the  perpendicular  let  fall  from  the  point  on  the 
plane.  If,  therefore,  on  two  planes  not  parallel  to  each  other,  whose  posi- 
tions are  known,  we  have  the  projections  of  a  point,  the  position  of  this 
point  is  completely  determined  by  erecting  perpendiculars  from  each  plane 
at  the  projected  points  :  their  intersection  will  be  the  point. 

If  from  every  point  of  an  indefinite  straight  line  A  B  (fig.  177),  placed 
in  any  manner  in  space,  perpendiculars  be  let 
fall  on  a  plane  L  M  N  O,  whose  position  is  given, 
then  all  the  points  in  which  these  perpendiculars 
meet  the  plane  will  form  another  indefinite  straight 
line  a  I :  this  line  is  called  the  projection  of  the 
line  AB  on  this  plane.  Since  two  points,  are 
sufficient  to  determine  a  straight  line,  it  is  only 
Fig.  177.  necessary  to  project  two  points  of  the  line,  and 

the  straight  line  drawn  through  the  two  projected  points  will  be  the  pro- 
jection of  the  given  line.  The  projection  of  a  straight  line,  itself  perpen- 
dicular to  the  plane,  is  the  point  in  which  this  perpendicular  meets  the 
plane. 

If  the  projections  a  b  and  a'  V  of  a  straight  line  on  the  two  planes 
L  M  K  O  and  L  M  P  Q  (fig.  178)  are  known,  this  line  A  B  is  determined ; 
for  if,  through  one  of  its  projections  a  5,  we  suppose  a  plane  drawn  perpen- 


GEOMETRICAL   PROJECTION. 


81 


dicularly  to  LMNO,  and  if  through  a'  V  another  plane  be  drawn  perpen- 
dicular to  L  M  P  Q,  the  intersection  of  the  two  planes  will  be  the  line  A  B. 
To  delineate  a  solid,  as  the  form  of  a  machine  for  instance,  it  must  be 
referred  to  three  series  of  dimensions,  each  of  them  at  right  angles  to  the 
plane  of  the  other.  g 


Fig.  ITS. 


Fig.  1T9. 


Thus,  let  a~b  G  (fig.  179)  be  a  parallelepiped  in  an  upright  position,  of 
which  the  plane  a  I  is  horizontal,  and  the  planes  a  c  and  c  I  vertical.  Let 
d  0,  df,  and  d  g,  be  the  boundary  planes  of  a  cubical  space  in  which  the 
body  a  T>  G  is  placed ;  the  sides  of  the  body  being  parallel  to  those  planes, 
each  to  each,  let  the  figure  of  the  parallelepiped  be  projected  on  these 
planes  ;  for  this  purpose  draw  parallel  lines  from  the  angles  of  the  body 
perpendicular  to  the  planes,  as  indicated  by  the  dotted  lines ;  then  upon  the 
plane  d  e  we  shall  have  a!  &',  the  projection  of  the  surface  a  I :  this  is  called 
the  plan  of  the  object.  Upon  the  plane  dfwe  have  a'  c',  the  projection 
of  the  surface  a  c,  the  front  elevation  /  and  upon  the  plane  dg,  the  projec- 
tion V  c'  of  the  surface  5  c,  the  side  elevation.  Here,  then,  we  have  three 
distinct  views  of  the  regular  solid  a  1)  c  delineated  on  plane  surfaces,  which 
convey  an  accurate  and  sufficient  idea  of  its  form.  Indeed,  any  two  of 
these  representations  are  sufficient  as  a  description  of  the  object.  From 
the  two  figures  a'  c',  V  c',  for  example,  the  third  figure  a'  V  may  be  com- 
pounded, by  merely  drawing  the  vertical  lines  c'  k,  V  i,  and  a!  k,  c'  I,  to 
meet  the  plane  d  e,  and  by  producing  them  horizontally  till  they  meet  and 


GEOMETRICAL   PKOJECTTON. 


m, a. 


form  the  figure  a'  V .  Similarly,  the  figure  V  c'  may  be  deduced  from  the 
other  two  by  the  aid  of  the  lines  h,  *,  from  a'  £',  and  the  lines  m,  n,  from 
a'  c'. 

It  is  in  this  way  that  a  third  view  of  any  piece  of  machinery  is  to  be 
found  from  two  given  views ;  and  in  many  cases  two  elevations,  or  one 
elevation  and  a  plan,  may  afford  a  sufficiently  complete  idea  of  the  con- 
struction of  a  machine.  In  other  cases,  many  parts  may  be  concealed  by 
others  in  which  they  are  enclosed ;  this  suggests  the  occasional  necessity 
of  views  of  the  interior,  in  which  the  machine  is  supposed  to  be  cut  across 
by  planes,  vertically  or  horizontally,  so  as  properly  to  reveal  its  structure. 
Such  views  are  termed  sections,  and,  with  reference  to  the  planes  of  section, 
are  denominated  vertical  and  horizontal  sections.  To  all  such  drawings 
is  given  the  general  title  of  geometrical  drawings,  as  distinguished  from 
perspective  drawings. 

By  the  aid  of  drawing  instruments,  measurements  are  transferable  from 

one  position  to  another  ;  and  there 
is  no  necessity  for  erecting  three 
such  planes  as  are  supposed  in  fig. 
179,  upon  which  to  execute  draw- 
ings of  a  machine.  In  practice, 
the  drawings  are  done  upon  one 
common  surface,  and  we  may  readi- 
ly suppose  the  plane  d  g  moved' 
back  into  the  position  d  g',  and  d  e 
also  moved  to  d  e',  both  of  these 
positions  being  in  the  plane  of  df. 
This  being  done,  we  have  the  three 
views  depicted  on  one  plane  surface 
(fig.  180).  In  this  figure,  the  same 
letters  of  reference  are  employed 
as  in  fig.  179 ;  d  I  and  d  m  are  the 
ground  and  vertical  lines.  It  is 
Fig-m  evident  that  the  positions  of  the 

same  points  in  a'  G'  and  a'  V  are  in  the  same  perpendicular  from  the  ground 
line  :  that,  in  short,  the  position  of  a  point  in  the  plane  may  be  found  by 
applying  the  edge  of  the  square  to  the  same  point  as  represented  in  the 
elevation.  The  same  remark  is  applicable  as  between  the  two  elevations. 
Hence  the  method  of  drawing  several  views  of  one  machine  upon  the  same 
surface  of  paper  in  strict  agreement  with  each  other. 


GEOMETRICAL   PROJECTION. 


83 


OF   SHADE   LINES. 

In  outline  drawings,  or  drawings  which  consist  simply  of  the  lines  em- 
ployed to  indicate  the  form  of  the  object  represented,  the  roundness,  the 
flatness,  or  the  obliquity  of  individual  surfaces,  is  not  indicated  by  the 
lines,  although  it  may  generally  be  inferred  from  the  relation  of  different 
views  of  the  same  part.  The  direct  significance  of  an  outline  drawing  may, 
however,  be  considerably  increased,  by  strengthening  those  lines  which  in- 
dicate the  contours  of  surfaces  resting  in  the  shadow ;  and  this  distinction 
also  improves  the  general  appearance  of  the  drawing.  The  strong  lines,  to 
produce  the  best  effect,  ought  to  be  laid  upon  the  sharp  edges  at  the  sum- 
mits of  salient  angles ;  but  bounding  lines  for  curve  surfaces  should  be 
drawn  finely,  and  should  be  but  slightly,  if  at  all,  strengthened  on  the 


Fig.  181. 


Fig.  183. 


Fig.  185. 


Fig.  182. 


Fig.  184. 


Fig.  186. 


shade  side.  This  distinction  assists  in  contrasting  flat  and  curve  surfaces. 
To  understand  and  apply  the  shade  lines,  however,  we  must  know  the 
direction  in  which  the  light  is  supposed  to  fall  upon  the  object,  and  thence 
the  locality  of  the  shadows. 

It  is  necessary  for  the  explicitness  of  the  drawing,  that  firstly,  the  light 
be  supposed  to  fall  upon  the  object  in  parallel  lines,  that  all  the  parts  may 


84  GEOMETRICAL   PKOJECTION. 

be  shade-lined  according  to  one  uniform  rule ;  secondly,  that  the  light 
should  be  supposed  to  fall  upon  the  object  obliquely,  as  in  this  way  both 
the  horizontal  and  vertical  lines  may  be  relieved  by  shading.  To  distribute 
the  shadows  equally,  the  light  is  supposed  to  fall  in  directions  forming  an 
angle  of  45°  with  both  the  horizontal  and  the  vertical  planes  of  projection. 
In  general,  the  light  should  fall,  as  it  were,  from  towards  the  upper  left- 
hand  corner  of  the  sheet  of  paper,  supposing  it  square,  making  also  an 
angle  of  45°  with  the  surface. 

,  To  illustrate  what  has  been  stated,  let  a  I  c  d  and  a'  I1  ef  (figs.  181  and 
182)  represent  the  elevation  and  plan  of  a  solid  rectangular  body,  N  O 
being  the  ground  line.  Let  the  direction  of  the  light  in  both  views  be 
represented  in  projection  by  the  arrows  A  B  ;  these  lines  form  the  angle 
45°  with  the  line  !N"  O,  and  by  drawing  the  parallels  at  J,  d,  a',  e',  so  as  to 
embrace  the  extreme  contour,  we  may  readily  perceive  the  way  in  which 
the  light  falls  upon  the  body :  it  falls  upon  three  faces,  namely,  the  two 
vertical  faces  a'f,fe,  and  the  top  a'  V  ef.  Consequently,  the  intersec- 
tions or  lines  at  which  these  planes  meet  ought  to  be  lightly  drawn,  namely, 
al,ad;  a'fandfe.  Again,  the  lateral  planes  represented  by  I  c,  cd, 
V  e',  and  a!  V,  are  obviously  in  the  shade,  as  no  light  falls  upon  them 
directly ;  and  these  lines  are  strengthened  to  express  the  distinction. 

In  figs.  183  and  184,  the  portion  of  the  exterior  from  I  by  c  to  d  is  in 
the  shade,  while  the  rest  is  light ;  and  the  inverse  is  the  case  with  the  inner 
edges.  A  peculiarity,  however,  occurs  at  d,  for  here  the  edges,  inner  and 
outer,  are  parallel  to  the  direction  of  the  light.  It  is  plain  that  the  surfaces 
which  come  up  to  these  edges  will  be  in  a  medium  shade,  and  that  the 
lines  at  d  should  be  of  medium  thickness. 

Figs.  185  and  186  represent  a  hollow  cylinder  in  projection.  In  the 
plan,  two  lines,  a,  c,  drawn  parallel  to  the  direction  of  the  light,  and  touch- 
ing the  exterior  of  the  cylinder,  define  the  semicircular  outline  a  V  c,  which 
is  thrown  in  the  shade,  and  ought  to  be  strengthened.  The  outlines  a  and 
c  are,  like  the  edges  at  d  (fig.  183),  parallel  to  the  light,  and  the  contour 
on  each  side  gradually  recedes  and  advances  to  the  light.  The  thickness 
of  the  line  should,  therefore,  be  rather  gradually  reduced  at  the  points  a,  c. 
In  the  elevation,  the  base-line  df  should  be  shaded,  and  b  d  is  often  half- 
shaded,  as  it  lies  in  a  curve  surface ;  more  generally  full-shaded. 

If,  again,  the  cylinder  be  hollow,  presenting  in  plan  the  interior  contour 
circle  e  h,  then  the  semicircle  e  g  li  expresses  the  shady  side  of  the  interior, 
the  light  striking  directly  upon  the  oppposite  semicircle. 

These  examples  illustrate  every  case  of  shade-lining  that  occurs  in  out- 
line drawings.  The  effect  is  enhanced  by  proportioning  the  thickness  of 


GEOMETRICAL   PROJECTION. 


85 


the  lines  to  the  depth  of  the  surfaces  to  which  they  belong,  below  the 

original  surfaces  from  which  the  shadows 

arise. 

In  the  later  French  system  of  shading, 
the  light  is  supposed,  in  plan,  to  strike  to- 
wards the  right  hand  upper  corner,  falling, 
as  it  were,  in  front  of  the  objects;  but  in 
elevation,  towards  the  right  hand  and  foot  of 
the  sheet  (figs.  188,  18T). 

It  will  be  observed  in  the  illustrations  of 
this  work,  that  in  the  tinted  drawings,  the 
shadow  is  thrown  according  to  the  French 
system  ;  that  is,  the  light  is  supposed  to  fall 
on  the  drawing  over  the  left  shoulder  at  an 
angle  of  45°.  But  in  outline  drawings,  on 
account  of  its  greater  simplicity,  the  more  rig.  m 

usual  system  of  throwing  the  shade  line  one  way,  both  in  plan  and  eleva- 
tion, is  adopted. 


PROJECTIONS   OF   SIMPLE  BODIES. 

Projections  of  a  regular  hexagonal  pyramid  (PI.  I,  II). — It  is  evident 
that  two  distinct  geometrical  views  are  necessary  to  convey  a  complete 
idea  of  the  form  of  the  object :  an  elevation  to  represent  the  sides  of  the 
body,  and  to  express  its  height ;  and  a  plan  of  the  upper  surface,  to  ex- 
press the  form  horizontally. 

It  is  to  be  observed  that  this  body  has  an  imaginary  axis  or  centre-line, 
about  which  the  same  parts  are  equally  distant ;  this  is  an  essential  charac- 
teristic of  all  symmetrical  figures. 

Draw  a  horizontal  straight  line  L  T  through  the  centre  of  the  sheet ; 
this  line  will  represent  the  ground  line.  Then  draw  a  perpendicular  Z  71 
to  the  ground  line.  For  the  sake  of  preserving  the  symmetry  of  the  draw- 
ing, the  centres  of  the  lower  range  of  figures  are  all  in  the  same  straight 
line  M  N,  drawn  parallel  to  the  ground  line. 

Figs.  1,  2. — In  delineating  the  pyramid,  it  is  necessary,  in  the  first 
place,  to  construct  the  plan.  The  point  S',  where  the  line  Z  71  intersects 
the  line  M  N",  is  to  be  taken  as  the  centre  of  the  figure,  and  from  this 
point,  with  a  radius  equal  to  the  side  of  the  hexagon  which  forms  the  base 
of  the  pyramid,  describe  a  circle,  cutting  M  N  in  A'  and  D'.  From  these 


86  GEOMETRICAL   PROJECTION. 

points  with  the  same  radius,  draw  four  arcs  of  circles,  cutting  the  primary 
circle  in  four  points.  These  six  points  being  joined  by  straight  lines,  will 
form  the  figure  A'  B'  0'  D'  E'  F',  which  is  the  base  of  the  pyramid  ;  and 
the  lines  A'  D',  B'  E',  and  C'  F',  will  represent  the  projections  of  its  edges 
fore-shortened  as  they  would  appear  in  the  plan.  If  this  operation  has 
been  correctly  performed,  the  opposite  sides  of  the  hexagon  should  be 
parallel  to  each  other  and  to  one  of  the  diagonals ;  this  should  be  tested 
by  the  application  of  the  square  or  other  instrument  proper  for  the  pur- 
pose. 

By  the  help  of  the  plan  obtained  as  above  described,  the  vertical  pro- 
jection of  the  pyramid  may  be  easily  constructed.  Since  its  base  rests 
upon  the  horizontal  plane,  it  must  be  projected  vertically  upon  the  ground 
line ;  therefore,  from  each  of  the  angles  at  A',  B',  C',  and  D',  raise  per- 
pendiculars to  that  line.  The  points  of  intersection,  A,  B,  C,  and  D,  are 
the  true  positions  of  all  the  angles  of  the  base ;  and  it  only  remains  to 
determine  the  height  of  the  pyramid,  which  is  to  be  set  off  from  the  point 
G  to  S,  and  to  draw  S  A,  S  B,  S  C,  and  S  D,  which  are  the  only  edges  of 
the  pyramid  visible  in  the  elevation.  Of  these  it  is  to  be  remarked  that 
S  A  and  S  D  alone,  being  parallel  to  the  vertical  plane,  are  seen  in  their 
true  length ;  and  moreover,  that  from  the  assumed  position  of  the  solid 
under  examination,  the  points  F'  and  E'  being  situated  in  the  lines  B  B' 
and  C  C',  the  lines  S  B  and  S  C  are  each  the  projections  of  two  edges  of 
the  pyramid. 

Figs.  3  and  4. — To  construct  the  projections  of  the  same  pyramid,  hav- 
ing its  base  set  in  an  inclined  position,  but  with  its  edges  S  A  and  S  D  still 
parallel  to  the  vertical  plane. 

It  is  evident,  that  with  the  exception  of  the  inclination,  the  vertical 
projection  of  this  solid  is  precisely  the  same  as  in  the  preceding  example, 
and  it  is  only  necessary  to  copy  fig.  1.  For  this  purpose,  after  having 
fixed  the  position  of  the  point  D  upon  the  ground  line,  draw  through  this 
point  a  straight  line  D  A,  making  with  L  T  an  angle  equal  to  the  desired 
inclination  of  the  base  of  the  pyramid.  Then  set  off  the  distance  D  A,  fig. 
1,  from  D  to  A,  fig.  3  ;  erect  a  perpendicular  on  the  centre,  and  set  off 
G  S  equal  to  the  height  of  the  pyramid.  Transfer  also  from  fig.  1  the  dis- 
tance B  G  and  C  G  to  the  corresponding  points  in  fig.  3,  and  complete  the 
figure  by  drawing  the  straight  lines  A  S,  B  S,  C  S,  and  D  S. 

In  constructing  the  plan  of  the  pyramid  in  this  position,  it  is  to  be  re- 
marked, that  since  the  edges  S  A  and  S  D  are  still  parallel  to  the  vertical 
plane,  and  the  point  D  remains  unaltered,  the  projection  of  the  point  A 
will  still  be  in  the  line  M  X.  Its  position  at  A'  (fig.  4)  is  determined  by 


GEOMETRICAL   PROJECTION.  87 

the  intersection  of  the  perpendicular  A  A'  with  that  line.  The  remaining 
points  B',  C',  &c.,  in  the  projection  of  the  base,  are  found  in  a  similar 
manner,  by  the  intersections  of  perpendiculars  let  fall  from  the  correspond- 
ing points  in  the  elevation,  with  lines  drawn  parallel  to  M  N",  at  a  distance 
(set  off  at  0,_p,)  equal  to  the  width  of  the  base.  By  joining  all  the  con- 
tiguous points,  we  obtain  the  figure  A'  B'  C'  D'  E'  F',  representing  the 
horizontal  projection  of  the  base,  two  of  its  sides,  however,  being  dotted, 
as  they  must  be  supposed  to  be  concealed  by  the  body  of  the  pyramid. 
The  vertex  S  having  been  similarly  projected  to  S',  and  joined  by  straight 
lines  to  the  several  angles  of  the  base,  the  projection  of  the  solid  is  com- 
pleted. •-•'.,  •; 

Figs.  5  and  6. — To  find  the  horizontal  projection  of  a  transverse  section 
of  the  same  pyramid,  made  by  a  plane  perpendicular  to  the  vertical,  but 
inclined  at  an  angle  to  the  horizontal  plane  of  projection  ;  and  let  oil  the 
sides  of  the  base  be  at  an  angle  with  the  ground  line. 

Having  drawn  the  vertical  S  S',  the  centre  line  of  the  figures,  its  point 
of  intersection  with  the  line  M  JN"  is  the  centre  of  the  plan.  Since  none  of 
the  sides  of  the  base  are  to  be  parallel  with  the  ground  line,  draw  a  diam- 
eter A'  D'  making  the  required  angle  with  that  line,  and  from  the  points 
A'  and  D'  proceed  to  set  out  the  angular  points  of  the  hexagon  as  in  fig.  2. 
Then,  in  order  to  obtain  the  projections  of  the  edges  of  the  pyramid,  join 
the  angular  points  which  are  diametrically  opposite  ;  and,  following  the 
method  pointed  out  in  reference  to  fig.  1,  project  the  figure  thus  obtained 
upon  the  vertical  plane,  as  shown  at  fig.  5. 

ISTow,  if  the  cutting  plane  be  represented  by  the  line  a  d  in  the  eleva- 
tion, it  is  obvious  that  it  will  expose,  as  the  section  of  the  pyramid,  a  poly- 
gon whose  angular  points  being  the  intersections  of  the  various  edges  with 
the  cutting  plane,  will  be  projected  in  perpendiculars  drawn  from  the 
points  where  it  meets  these  edges  respectively.  If,  therefore,  from  the 
points  a,  /,  b,  &c.,  we  let  fall  the  perpendiculars  a  a',  ff,  b  b',  &c.,  and 
join  their  contiguous  points  of  intersection  with  the  lines  A'  D',  F'  C',  B'  E', 
&c.,  we  shall  form  a  six-sided  figure,  which  will  represent  the  section  re- 
quired. The  edges  F  S  and  E  S  being  concealed  in  the  elevation,  but 
necessary  for  the  construction  of  the  plan,  have  been  expressed  in  dotted 
lines,  as  also  the  portion  of  the  pyramid  situated  above  the  cutting  plane, 
which,  though  supposed  to  be  removed,  is  necessary  in  order  to  draw  the 
lines  representing  the  edges.  "VVe  have  here  introduced  the  ordinary 
method  of  expressing  sections  in  purely  line-drawings,  by  filling  up  the 
spaces  comprised  within  their  outlines  with  a  quantity  of  parallel  straight 
lines  drawn  at  equal  distances. 


88  GEOMETRICAL   PROJECTION. 

Figs.  7  and  8. — To  find  the  horizontal  projection  of  the  transverse  sec- 
tion of  a  regular  five-sided  pyramid,  cut  by  a  plane  perpendicular  to  the 
vertical,  but  inclined  to  the  horizontal  plane  ;  and  let  one  edge  of  the  pyra- 
mid he  in  a  plane  perpendicular  to  both  planes  of  projection. 

The  plan  of  the  pyramid  is  constructed  by  describing  from  the  centre 
S'  a  circle  circumscribing  the  base,  and  from  B'  dividing  the  circumference 
into  five  equal  parts,  and  joining  the  contiguous  points  of  division  by 
straight  lines  to  form  the  polygon  A'  B'  C'  D'  E',  each  of  whose  angles, 
being  joined  to  the  centre  S',  shows  the  projections  of  the  edges  of  the 
pyramid.  Then,  following  the  method  above  explained,  we  obtain  the 
elevation  and  the  horizontal  projection  of  the  section  made  by  the  plane 
a  c.  But  that  method  will  not  suffice  for  the  determination  of  the  point  V, 
because  the  perpendicular  let  fall  from  the  corresponding  point  5,  in  the 
elevation,  coincides  with  the  projection  of  the  edge  B  S.  Let  the  pyramid 
be  supposed  to  be  turned  a  quarter  of  a  revolution  round  its  axis  ;  the  line 
B'  S'  will  then  have  assumed  the  position  S'  J4.  Project  the  point  V  to  5s, 
and  join  S  V.  Then,  since  the  required  point  must  also  be  conceived  to 
have  described  a  quarter  of  a  circle  in  a  plane  parallel  to  the  horizontal 
plane,  and  that  its  new  position  must  be  in  the  line  S  5s,  it  is  obvious  that 
its  vertical  projection  is  the  point  J4,  the  intersection  of  a  horizontal  line 
drawn  through  5,  with  that  line.  The  distance  5  £4,  then,  being  transferred 
from  S'  to  V ,  determines  the  position  of  the  latter  point  in  the  plan ;  or, 
following  a  more  methodical  process,  by  projecting  the  point  J4  to  V1,  and 
describing  a  circle  from  the  centre  S'  passing  through  V ;  its  intersection 
with  B'  S'  is  the  point  sought. 

PROJECTIONS   OF  A   PRISM. 

Plate  III.,  figs.  1  and  2. — Required  to  represent  in  plan  and  elevation 
a  regular  six-sided  prism  in  an  upright  position. 

Lay  down  the  ground  line  and  centre  line,  and  describe  the  hexagon  as 
already  directed.  Project  the  plan  thus  delineated  by  perpendiculars  to 
the  ground  line  from  each  of  its  angular  points  ;  and  since  the  prism  is 
upright,  these  angular  points  themselves  represent  the  horizontal  projec- 
tions of  all  its  edges,  and  their  elevations  coincide  with  the  perpendiculars 
A'  G,  B'  H,  &c.  Set  off  from  G  to  A  the  height  of  the  prism,  and  through 
A  draw  A  D  parallel  to  the  ground  line.  This  will  be  the  vertical  projec- 
tion of  the  upper  surface.  The  edges  being  all  parallel  to  the  vertical 
plane,  are,  of  course,  seen  in  their  actual  length. 

Figs.  3  and  4. — To  form  the  projections  of  the,  same  prism,  supposing  it 


GEOMETRICAL   PROJECTION.  89 

to  have  been  moved  round  the  point  G,  in  a  plane  parallel  to  the  vertical 
plane. 

Copy  the  elevation  (fig.  1)  on  an  inclined  base.  Now,  by  letting  fall 
perpendiculars  from  all  the  angles  in  the  elevation,  and  joining  the  con- 
tiguous points  of  intersection  with  the  horizontal  lines  appropriate  to  these 
points  respectively,  we  obtain  the  polygon  A'  B'  C'  D7  E7  F'  as  the  projec- 
tion of  the  upper  surface,  and  G'  H'  I7  K'  I/M7  as  that  of  the  base  of  the 
prism.  Finally,  it  will  be  observed  that  all  the  edges  are  represented,  in 
the  horizontal  projection,  by  equal  straight  lines,  as  D'  K7,  A7  G',  &c.,  and 
that  the  sides  A'  B',  G7  H',  &c.,  remain  still  parallel  to  each  other,  which 
will  afford  the  means  of  verifying  the  accuracy  of  the  drawings. 

Figs.  5  and  6. — Required  the  projections  of  the  same  prism  set  into  a 
position  inclined  to  both  planes  of  projection. 

Assuming  that  the  inclination  of  the  prism  upon  the  horizontal  plane 
is  the  same  as  in  the  preceding  figures  for  the  sake  of  simplifying  the  opera- 
tion, the  first  process  is  to  copy  fig.  4,  which  may  be  done  by  drawing  a 
centre  line  X  X,  so  as  to  form  the  required  angle  of  the  prism  with  the 
vertical  plane  ;  then,  having  set  off  upon  this  line  a  distance  equal  to 
A7  K',  fig.  4,  transfer  the  distances  A7  G'  and  D7  K7  also  to  fig.  6  ;  and  in 
order  to  find  the  remaining  angular  points,  make  A7  a  equal  to  the  corre- 
sponding distance  in  fig.  4,  and  through  a  draw  B7  F7  perpendicular  to  the 
centre  line,  and  transfer  the  distances  a  B7,  a  F7.  Through  the  points  B7 
and  F7,  draw  straight  lines  parallel  to  A7  K7,  and  join  A7  B7,  A7  F7 ;  and 
since  we  have  already  seen  that  all  the  other  sides  must  be  parallel  to  these, 
the  figure  is  completed  by  drawing  through  the  points  G7,  D7,  and  K7, 
straight  lines  parallel  to  A7  B7  and  A7  F7  respectively. 

Now,  since  the  prism  has  been  supposed  to  have  preserved  its  former 
inclination  to  the  horizontal  plane,  it  is  obvious  that  every  point  in  it,  such 
as  A,  has,  in  assuming  its  new  position,  simply  moved  in  a  horizontal 
plane,  and  will,  therefore,  be  in  the  line  A  A  parallel  to  the  ground  line, 
and  since  the  same  point  has  been  projected  to  A7,  fig.  6,  it  will  also  be  in 
the/  perpendicular  A!  A  ;  the  point  of  intersection  A,  fig.  5,  is,  therefore, 
its  projection  in  the  elevation.  The  remaining  angular  points  in  this  view 
are  all  determined  in  the  same  manner  by  the  aid  of  figs.  3  and  6  ;  and 
having  joined  the  contiguous  points,  and  the  corresponding  angles  of  the 
upper  and  lower  surface,  we  obtain  the  complete  vertical  projection  of  the 
prism  in  its  doubly-inclined  position. 


90  GEOMETRICAL   PROJECTION. 


CONSTRUCTION   OF  THE   CONIC    SECTIONS. 

Plate  V. — The  plan  of  the  cone  (fig.  2)  is  simply  a  circle,  described 
from  the  centre  S'  of  a  diameter  equal  to  that  of  the  base.  Its  elevation 
(fig.  1)  is  an  isosceles  triangle,  obtained  by  drawing  tangents  A'  A,  B'  B, 
perpendicular  to  and  intersecting  the  ground  line ;  then  set  off  upon  the 
centre  line  the  height  C  S,  and  join  S  A,  S  B.  These  lines  are  called  the 
exterior  generatrices  of  the  cone. 

Figs.  1  and  2. — Given  tlie  projections  of  a,  cone,  and  the  direction  of  a 
plane  X  X,  cutting  it  perpendicularly  to  the  vertical,  and  obliquely  to  the 
horizontal  plane  ;  required  to  find,  first,  the  horizontal  projection  of  this 
section  •  and,  secondly,  the  outline  of  the  ellipse  thus  formed. 

Through  the  vertex  of  the  cone  draw  a  line  S  E  to  any  point  within  the 
base  A  B ;  let  fall  a  perpendicular  from  E,  cutting  the  circumference  of 
the  base  in  E',  and  join  E'  S' ;  then  another  perpendicular  let  fall  from  e 
will  intersect  E7  S'  in  a  point  e' ',  which  will  be  the  horizontal  projection 
of  a  point  in  the  curve  required  ;  and  so  on  for  any  required  number  of 
points. 

The  exterior  generatrices  A  S  and  B  S  being  both  projected  upon  the 
line  A'  B',  the  extreme  limits  of  the  curve  sought  will  be  at  the  points  a' 
and  y  on  that  line,  which  are  the  projections  of  the  points  of  intersection 
a  and  5  of  the  cutting  plane  with  the  outlines  of  the  cone.  And  since  the 
line  a'  V  will  obviously  divide  the  curve  symmetrically  into  two  equal 
parts,  the  points  /',  g' ,  h',  &c.,  will  be  readily  obtained  by  setting  off 
above  that  line,  and  on  their  respective  perpendiculars,  the  distances  d'  d?, 
e'  e\  &c.  A  sufficient  number  of  points  having  thus  been  determined,  the 
curve  drawn  through  them  (which  will  be  found  to  be  an  ellipse)  will  be 
the  outline  of  the  section  required. 

This  curve  may  be  obtained  by  another  method,  depending  on  the  prin- 
ciple that  all  sections  of  a  cone  by  planes  parallel  to  the  base  are  circles. 
Thus,  let  the  line  F  G  represent  a  cutting  plane ;  the  section  which  it 
makes  with  the  cone  will  be  denoted  on  the  horizontal  projection  by  a 
circle  drawn  from  the  centre  S',  with  a  radius  equal  to  half  the  line  F  G ; 
and  by  projecting  the  point  of  intersection  II  of  the  horizontal  and  oblique 
planes  by  a  perpendicular  H  H',  and  noting  where  this  line  cuts  the  circle 
above  referred  to,  we  obtain  two  points  IT  and  I'  in  the  curve  required. 
By  a  similar  construction,  as  exemplified  in  the  drawings,  any  number  of 
additional  points  may  be  found. 

As  the  projection  obtained  by  the  preceding  methods  exhibits  the  sec- 


GEOMETRICAL   PROJECTION.  91 

tion  as  fore-shortened,  and  not  in  its  true  dimensions,  we  shall  now  proceed 
to  the  consideration  of  the  second  question  proposed.  Let  the  cutting 
plane  X  X  be  conceived  to  turn  upon  the  point  b,  so  as  to  coincide  with 
the  vertical  line  b  &,  and  (to  avoid  confusion  of  lines)  let  b  k  be  transferred 
to  a'  b',  which  will  represent,  as  before,  the  extreme  limits  of  the  curve 
required.  Now,  taking  any  point,  such  as  d,  it  is  obvious  that  in  this  new 
position  of  the  cutting  plane,  it  will  be  represented  by  d*,  and  if  the  cutting 
plane  were  turned  upon  a!  V  as  an  axis  till  it  is  parallel  to  the  vertical 
plane,  the  point  which  had  been  projected  at  d*  would  then  have  described 
round  a'  V  an  arc  of  a  circle,  whose  radius  is  the  distance  d?  d?  (fig.  2). 
This  distance,  therefore,  being  set  off  at  d'  and  f  on  each  side  of  a'  b' ', 
gives  two  points  in  the  curve  sought.  By  a  similar  mode  of  operation  any 
number  of  points  may  be  obtained,  through  which,  if  a  curve  be  drawn,  it 
will  be  an  ellipse  of  the  true  form  and  dimensions  of  the  section. 

Figs.  3  and  4. — To  find  the  horizontal  projection  and  actual  oiiiline  of 
the  section  of  a  cone,  made  by  a  plane  Y  Y  parallel  to  one  side  or  generatrix, 
and  perpendicular  to  the  vertical  plane. 

Determine  by  the  second  method  laid  down  in  the  preceding  problem 
any  number  of  points,  as  F',  G',  J',  K',  &c.,  in  the  curve  representing  the 
horizontal  projection  of  the  section  specified.  The  horizontal  plane  pass- 
ing through  M  gives  only  one  point  M'  (which  is  the  vertex  of  the  curve 
sought),  because  the  circle  which  denotes  the  section  that  it  makes  with 
the  cone  is  a  tangent  to  the  given  plane. 

In  order  to  determine  the  actual  outline  of  this  curve,  suppose  the 
plane  Y  Y  to  turn  as  upon  a  pivot  at  M,  until  it  has  assumed  the  position 
M  B,  and  transfer  M  B  parallel  to  itself  to  M'  B'.  The  point  F  will  thus 
have  first  described  the  arc  F  E  till  it  reaches  the  point  E,  which  is  then 
projected  to  E2;  suppose  the  given  plane,  now  represented  by  M'  B',  to 
turn  upon  that  line  as  an  axis,  until  it  assumes  a  position  parallel  to  the 
vertical  plane,  the  point  E2,  which  is  distant  from  the  axis  M'  B'  by  the 
distance  F'  S'  (fig.  4),  will  now  be  projected  to  F'  (fig.  3).  The  same  dis- 
tance F'  S'  set  off  on  the  other  side  of  the  axis  H'  B'  gives  another  point 
G'  in  the  curve  required,  which  is  the  parabola. 

Figs.  7,  8,  9. — To  draw  the  vertical  projection  of  the  sections  of  two  op- 
posite cones  made  by  a  plane  parallel  to  their  axis. 

Let  C  E  D  and  C  B  A  be  the  two  cones,  and  X  X  the  position  of  the 
cutting  plane  (fig.  7).  Project  in  plan  either  of  the  cones,  as  in  fig.  8  ; 
from  its  centre,  with  a  radius  equal  to  L  H,  describe  a  circle,  and  draw 
the  tangent  b  a  ;  ba  will  be  the  horizontal  projection  of  the  cutting  plane. 
Draw  the  line  II'  M'  (fig.  9)  parallel  to  the  cutting  plane ;  H',  M'  corre- 


92  GEOMETRICAL   PROJECTION. 

spending  in  position  to  the  intersections  H,  M  (fig.  7"),  of  the  plane  with  the 
cones.  From  H'  and  M'  lay  off  distances  equal  to  L  K,  K  I,  and  the  length  of 
the  cone,  and  through  these  points  draw  perpendiculars,  as/'  <?',  d'  c',  V  a', 
&c.,  which  must  be  made  equ^l  to  the  chords/ 0,  dc^ba  (fig.  8),  made  by 
the  cutting  plane  a  5,  with  circles  whose  radii  are  G K,  IF,  and  the  radius 
of  the  base  of  the  cone.  Through  the  points  a',  c',  e',  H',/',  d',  V,  draw 
the  curve,  and  we  have  the  projection  required.  A  similar  construction 
will  give  the  sectional  projection  of  the  opposite  cone  at  W.  The  curve 
thus  found  is  the  hyperbola. 


PENETRATIONS   OR   INTERSECTIONS   OF   SOLIDS. 

On  examining  the  minor  details  of  most  machines,  we  find  numerous 
examples  of  cylindrical  and  other  forms,  fitted  to,  and  even  appearing  to 
pass  through  each  other  in  a  great  variety  of  ways.  The  examples  grouped 
in  plates  VI.  to  XI.  are  selected  with  the  view  of  exhibiting  those  cases 
which  are  of  most  frequent  occurrence,  and  of  elucidating  general  principles. 


PENETRATIONS    OF   CYLINDERS. 

Plate  VI. — Figs.  1  and  2  represent  the  projections  of  two  cylinders  of 
unequal  diameters  meeting  each  other  at  right  angles ;  one  of  which  is 
denoted  by  the  rectangle  A  B  E  D  in  the  vertical,  and  by  the  circle 
A!  H'  B'  in  the  horizontal  projections  ;  while  the  other,  which  is  supposed 
to  be  horizontal,  is  indicated  in  the  former  by  the  circle  L  P  I  !N",  and  in 
the  latter  by  the  figure  L'  I'  K'  M'.  From  the  position  of  these  two  solids 
it  is  evident  that  the  curves  formed  by  their  junction  will  be  projected  in 
the  circles  A'  H'  B'  and  L  P  I  !N" ;  and  further,  that  such  would  also  be 
the  case  even  although  their  axes  did  not  intersect  each  other. 

But  if  the  position  of  these  bodies  be  changed  into  that  represented  at 
figs.  3  and  4,  the  lines  of  their  intersection  will  assume  in  the  vertical  pro- 
jection a  totally  different  aspect,  and  may  be  accurately  determined  by 
the  following  construction. 

Through  any  point  taken  upon  the  plan  (fig.  4)  draw  a  horizontal  line 
a'  I', which  is  to  be  considered  as  indicating  a  plane  cutting  both  cylinders 
parallel  to  their  axes  ;  this  plane  would  cut  the  vertical  cylinder  in  lines 
drawn  perpendicularly  through  the  points  c'  and  d'.  To  find  the  vertical 
projection  of  its  intersection  with  the  other  cylinder,  conceive  its  base  I'  L', 


GEOMETRICAL   PROJECTION.  93 

after  being  transferred  to  P  L3,  to  be  turned  over  parallel  to  the  horizontal 
plane  ;  this  is  expressed  by  simply  drawing  a  circle  of  the  diameter  I2  L2 ; 
and  producing  the  line  a'  b'  to  a2 ;  then  set  off  the  distance  a*  e'  on  each 
side  of  the  axis  I  K,  and  draw  straight  lines  through  these  points  parallel 
to  it.  These  lines  a  &,  g  A,  denote  the  intersection  of  the  plane  a!  b'  with 
the  horizontal  cylinder,  and  therefore  the  points  c,  d,  m,  o,  where  they  cut 
the  perpendiculars  c  c',  d  d',  are  points  in  the  curve  required.  By  laying 
down  other  planes  similar  to  a!  I',  and  operating  as  before,  any  number  of 
points  may  be  obtained.  The  vertices  i  and  k  of  the  curves  are  obviously 
projected  directly  ;  and  their  extreme  points  are  determined  by  the  inter- 
sections of  the  outlines  of  both  cylinders.  When  the  cylinders  are  of 
unequal  diameters,  as  in  the  present  case,  the  curves  of  penetration  are 
hyperbolas. 

Figs.  5  and  6. —  When  the  diameters  of  the  cylinders  are  equal,  and 
when  they  cut  each  other  at  right  angles,  the  curves  of  penetration  are  pro- 
jected vertically  in  straight  lines  perpendicular  to  each  other,  as  in  fig.  5, 
where  the  projections  of  some  of  the  points  are  indicated  in  elevation  and 
plan  by  the  same  letters  of  reference. 

Figs.  7  and  8. — To  delineate  the  intersections  of  two  cylinders  of  equal 
diameters  at  right  angles,  when  one  of  the  cylinders  is  inclined  to  the  ver- 
tical plane. 

Supposing  the  two  preceding  figures  to  have  been  drawn,  the  projec- 
tion c  of  any  point  such  as  c'  may  be  ascertained  by  observing  that  it  must 
be  situated  in  the  perpendicular  c'  <?,  and  that  since  the  distance  of  this 
point  (projected  at  c  in  fig.  5)  from  the  horizontal  plane  remains  unaltered, 
it  must  also  be  in  the  horizontal  line  c  c.  Upon  these  principles  all  the 
points  indicated  by  literal  references  in  fig.  7  are  determined  ;  the  curves 
of  penetration  resulting  therefrom  intersecting  each  other  at  two  points 
projected  upon  the  axial  line  L  K,  of  which  that  marked  q  alone  is  seen. 
The  ends  of  the  horizontal  cylinder  are  represented  by  ellipses,  the  con- 
struction of  which  will  also  be  obvious  on  referring  to  the  figures ;  and 
they  do  not  require  further  consideration  here. 


PENETRATIONS   OF   CYLINDERS,    CONES,   AND   SPHERES. 

PL  YIIL,  figs.  1  and  2. — To  find  the  curves  resulting  from  the  inter- 
section of  two  cylinders  of  unequal  diameters,  meeting  at  any  angle. 

For  the  sake  of  simplicity,  suppose  the  axes  of  both  cylinders  to  be 
parallel  to  the  vertical  plane,  and  let  A  B  E  D  and  N  O  Q  P  be  their  pro- 


94;  GEOMETRICAL   PROJECTION. 

jections  upon  that  plane.  In  constructing,  in  the  first  place,  their  horizon- 
tal projection,  observe  that  the  upper  end  A  B  of  the  larger  cylinder  is 
represented  by  an  ellipse  A'  K'  B'  M',  which  may  easily  be  drawn  by  the 
help  of  the  major  axis  K'  M'  equal  to  the  diameter  of  the  cylinder,  and  of 
the  minor  A'  B',  the  projection  of  the  diameter.  The  visible  portion  of 
the  base  of  the  cylinder  being  similarly  represented  by  the  semi-ellipse 
I/  D'  H',  its  entire  outline  will  be  completed  by  drawing  tangents  I/  M' 
and  IF  E7.  The  upper  extremity  P  K  of  the  smaller  cylinder  will  also  be 
projected  in  the  ellipse^'  i'  W. 

~R ow,  suppose  a  plane,  as  a'  g'  (fig.  2),  to  pass  through  both  cylinders 
parallel  to  their  axes  ;  it  will  cut  the  surface  of  the  larger  cylinder  in  two 
straight  lines  passing  through  the  points  f  and  g'  on  the  upper  end  of  the 
cylinder ;  these  lines  will  be  represented  in  the  elevation,  by  projecting 
the  points  y  and  g'  to/*,  g  j  and  drawing  af  and  c  g  parallel  to  the  axis. 
The  plane  af  g'  will  in  like  manner  cut  the  smaller  cylinder  in  two  straight 
lines,  which  will  be  represented  in  the  vertical  projection  by  d  h  and  e  i, 
and  the  intersections  of  these  lines  with  af  and  c  g  will  give  four  points 
I,  k,  ra,  and  n,  in  the  curves  of  penetration.  Of  these  points  one  only,  that 
marked  ?,  is  visible  in  the  plan,  where  it  is  denoted  by  I'. 

Fig.  1. — To  find  the  curves  of  penetration  in  the  elevation  without  the 
aid  of  the  plan. 

Let  the  bases  D  E  and  Q  O  of  both  cylinders  be  conceived  to  be  turned 
over  into  the  vertical  plane  after  being  transferred  to  any  convenient  dis- 
tance, as  D2  E2  and  Q2  O2,  from  the  principal  figure  ;  they  will  then  be 
represented  by  the  circles  D2  H2  E2  and  Q2  G'  O2.  Now  draw  a*c*  paral- 
lel to  D  E,  and  at  any  suitable  distance  from  the  centre  I ;  this  line  will 
represent  the  intersection  of  the  base  of  the  cylinder  with  a  plane  parallel 
to  the  axes  of  both,  as  before.  The  intersection  of  this  plane  with  the 
base  of  the  smaller  cylinder  will  be  found  by  setting  off  from  E,  a  distance 
Rj?,  equal  to  I  o,  and  drawing  through  the  point  p  a  straight  line  parallel 
to  Q  O.  It  is  obvious  that  the  intersection  of  the  supposed  plane  with  the 
convex  surfaces  of  the  cylinders  will  be  represented  by  the  lines  af,  c  g, 
and  d  A,  e  i,  drawn  parallel  to  the  axes  of  the  respective  cylinders  through 
the  points  where  the  chords  a*  cz  and  d2  ez  cut  the  circles  of  their  bases  ; 
and  that,  consequently,  the  intersections  of  these  lines  indicate  points  in 
the  curves  sought.  These  points  may  be  multiplied  indefinitely  by  con- 
ceiving other  planes  to  pass  through  the  cylinders,  and  operating  as 
before. 

Eigs.  3  and  4.  —  To  find  the  curves  of  penetration  of  a  cone  and, 


GEOMETRICAL   PROJECTION.  95 

Let  D  S  be  the  axis  of  the  cone,  A'  I/  B'  the  circle  of  its  base,  and  the 
triangle  A  B  S  its  projection  on  the  vertical  plane  ;  and  let  C,  C',  be  the 
projections  of  the  centre,  and  the  circles  E'  K7  F'  and  E  G  F  those  of  the 
circumferences  of  the  sphere. 

This  problem,  like  most  others  similar  to  it,  can  be  solved  only  by  the 
aid  of  imaginary  intersecting  planes.  Let  a  b  (fig.  3)  represent  the  pro- 
jection of  a  horizontal  plane  ;  it  will  cut  the  sphere  in  a  circle  whose  diam- 
eter is  a  T),  and  which  is  to  be  drawn  from  the  centre  C'  in  the  plan. 
Its  intersection  with  the  cone  is  also  a  circle  described  from  the  centre  S' 
with  the  diameter  c  d;  the  points  e'  and/',  where  these  two  circles  cut 
each  other,  are  the  horizontal  projections  of  two  points  in  the  lower  curve, 
which  is  evidently  entirely  hidden  by  the  sphere.  The  points  referred  to 
are  projected  vertically  upon  the  line  a  5  at  e  and  f.  The  upper  curve, 
which  is  seen  in  both  projections,  is  obtained  by  a  similar  process  ;  but  it 
is  to  be  observed  that  the  horizontal  cutting  planes  must  be  taken  in  such 
positions  as  to  pass  through  both  solids  in  circles  which  shall  intersect  each 
other.  For  our  guidance  in  this  respect  it  will  be  necessary,  first,  to  de- 
termine the  vertices  m  and  n  of  the  curves  of  penetration. 

For  this  purpose,  conceive  a  vertical  plane  passing  through  the  axis  of 
the  cone  and  the  centre  of  the  sphere ;  its  horizontal  projection  will  be  the 
straight  line  G'  L'  joining  the  centres  of  the  two  bodies.  Let  us  also  make 
the  supposition  that  this  plane  is  turned  upon  the  line  C  C'  as  on  an  axis, 
until  it  becomes  parallel  to  the  vertical  plane ;  the  points  S'  and  L'  will 
now  have  assumed  the  positions  S2  and  L2,  and  consequently  the  axis  of 
the  cone  will  be  projected  vertically  in  the  line  D'  S3,  and  its  side  in  S3  L3, 
cutting  the  sphere  at  the  points  p  and  r.  Conceive  the  solids  to  have 
resumed  their  original  relative  positions,  it  is  clear  that  the  vertices  or 
adjacent  limiting  points  of  the  curves  of  penetration  must  be  in  the  hori- 
zontal lines  p  o  and  r  q,  drawn  through  the  points  determined  as  above  ; 
their  exact  positions  on  these  lines  may  be  ascertained  by  projecting  ver- 
tically the  points  ra'  and  n't  where  the  arcs  described  by  the  points  p  and 
r,  in  restoring  the  cone  to  its  first  position,  intersect  the  line  S  L. 

It  is  of  importance  further,  to  ascertain  the  points  at  which  the  curves 
of  penetration  meet  the  outlines  A  S  and  S  B  of  the  cone.  The  plane 
which  passes  through  these  lines  being  projected  horizontally  in  A'  B',  will 
cut  the  sphere  in  a  circle  whose  diameter  is  i'j'j  this  circle,  described  in  the 
elevation  from  the  centre  C,  will  cut  the  sides  A  S  and  S  B  in  four  points 
at  which  the  curves  of  penetration  are  tangents  to  the  outlines  of  the  cone. 

Figs.  5  and  6. — To  find  the  lines  of  penetration  of  a  cylinder  and  a 
cylindrical  ring  or  torus. 


96  GEOMETRICAL   PROJECTION. 

Let  the  circles  A'  E'  B',  F'  G'  K/,  represent  the  horizontal,  and  the 
figure  A  C  B  D  the  vertical  projection  of  the  torus,  and  let  the  circle 
H'/'L',  and  the  rectangle  H  I  M  L  be  the  analogous  projections  of  the 
cylinder,  which  passes  perpendicularly  through  it.  Conceive,  as  before, 
a  plane  a  ~b  (fig.  5),  to  pass  horizontally  through  both  solids ;  it  will  ob- 
viously cut  the  cylinder  in  a  circle  which  will  be  projected  in  the  base 
IT  f  I/  itself,  and  the  ring  in  two  other  circles,  of  which  one  only,  part 
of  which  is  represented  by  the  arc  f  b3  b',  will  intersect  the  cylinder  at 
the  points  f  and  J3,  which  being  projected  vertically  to  fig.  5,  will  give 
two  points/1  and  5*  in  the  upper  curve  of  penetration. 

Another  horizontal  plane,  taken  at  the  same  distance  below  the  centre 
line  A  B  as  that  marked  a  b  is  above  it,  will  evidently  cut  the  ring  in 
circles  coinciding  with  those  already  obtained  ;  consequently  the  points/' 
and  b3  indicate  points  in  the  lower  as  well  as  in  the  upper  curves  of  pene- 
tration, and  are  projected  vertically  at  d  and  e.  Thus,  by  laying  down 
two  planes  at  equal  distances  on  each  side  of  A  B,  by  one  operation  four 
points  in  the  curves  required  are  determined. 

To  determine  the  vertices  m  and  n,  following  the  method  explained  in 
the  preceding  problem,  draw  a  plane  O  ri ',  passing  through  the  axis  of  the 
cylinder  and  the  centre  of  the  ring,  and  conceive  this  plane  to  be  moved 
round  the  point  O  as  on  a  hinge,  until  it  has  assumed  the  position  O  B', 
parallel  to  the  vertical  plane ;  the  point  n',  representing  the  extreme 
outline  of  the  cylinder  in  plan,  will  now  be  at  r',  and  being  projected  ver- 
tically, that  outline  will  cut  the  ring  in  two  points  p  and  r,  which  would 
be  the  limits  of  the  curves  of  penetration  in  the  supposed  relative  position 
of  the  two  solids  ;  and  by  drawing  the  two  horizontal  lines  r  n  and  p  m, 
and  projecting  the  point  n'  vertically,  the  intersections  of  these  lines,  the 
two  points,  m  and  n,  are  the  vertices  of  the  curves  in  the  actual  position  of 
the  penetrating  bodies. 

The  points  at  which  the  curves  are  tangents  to  the  outlines  II I  and  L  M 
of  the  cylinder,  may  readily  be  found  by  describing  arcs  of  circles  from 
the  centre  O  through  the  points  H'  and  L',  which  represent  these  lines  in 
the  plan,  and  then  proceeding,  as  above,  to  project  the  points  thus  obtained 
upon  the  elevation.  Lastly,  to  determine  the  points,  as  j,  z,  &c.,  where 
the  curves  are  tangents  to  the  horizontal  outlines  of  the  ring,  draw  a  circle 
P'  B'  j'  with  a  radius  equal  to  that  of  the  centre  line  of  the  ring,  namely, 
P  D  ;  the  points  of  intersection  d  and  j'  are  the  horizontal  projections  of 
the  points  sought. 

Required  to  represent  the  sections  which  would  be  made  in  the  ring  now 
before  us,  by  two  planes,  one  of  which,  N'  T',  is  parallel  to  the  vertical 


GEOMETRICAL   PROJECTION.  97 

plane,  while  the  other,  T"  E',  is  perpendicular  to  both  planes  of  projec- 
tion. 

The  section  made  by  the  last-named  plane  must  obviously  have  its  ver- 
tical projection  in  the  line  C  D,  which  indicates  the  position  of  the  plane ; 
but  the  former  will  be  represented  in  its  actual  form  and  dimensions  in 
the  elevation.  To  determine  its  outlines,  let  two  horizontal  planes  g  q  and 
i  k,  equidistant  from  the  centre  line  A  B,  be  supposed  to  cut  the  ring ; 
their  lines  of  intersection  with  it  will  have  their  horizontal  projections  in 
the  two  circles  g'  o'  and  hf  q'  which  cut  the  given  plane  N'  T'  in  o'  and  qf. 
These  points  being  projected  vertically  to  o,  q,  k,  &c.,  give  four  points  in 
the  curve  required.  The  line  N'  T'  cutting  the  circle  A'  E'  B'  at  1ST',  the 
projection  N"  of  this  point  is  the  extreme  limit  of  the  curve. 

The  circle  P'  s' f,  the  centre  line  of  the  rim  of  the  torus,  is  cut  by  the 
planes  N'  T'  at  the  point  s',  which  being  projected  vertically  upon  the 
lines  D  P  and  C  I,  determines  s  and  I,  the  points  of  contact  of  the  curve 
with  the  horizontal  outlines  of  the  ring.  Finally,  the  points  t  and  u  are 
obtained  by  drawing  from  the  centre  O  a  circle  T'  vf  tangent  to  the  given 
plane,  and  projecting  the 'point  of  intersection  vf  to  the  points  v  and  a?, 
which  are  then  to  be  replaced  upon  C  D  by  drawing  the  horizontals  v  t 
and  x  u. 


PENETRATIONS   OF  CYLINDERS,    PRISMS,    SPHERES,   AND   CONES. 

Plate  X.,  figs.  1  and  2. — Required  to  delineate  the  lines  of  penetra- 
tion of  a  sphere  and  a  regular  hexagonal  prism  whose  axis  passes  through 
the  centre  of  the  sphere. 

The  centres  of  the  circles  forming  the  two  projections  of  the  sphere  are, 
according  to  the  terms  of  the  problem,  upon  the  axis  C  C'  of  the  upright 
prism,  which  is  projected  horizontally  in  the  regular  hexagon  D'  E'  F'  G' 
IF  I'.  Hence  it  follows,  that  as  all  the  lateral  faces  of  the  prism  are  equi- 
distant from  the  centre  of  the  sphere,  their  lines  of  intersection  with  it  will 
necessarily  be  circles  of  equal  diameters.  Now,  the  perpendicular  face 
represented  by  the  line  E'  F'  in  the  plan,  will  meet  the  surface  of  the 
sphere  in  two  circular  arcs  E  F  and  L  M  (fig.  1),  described  from  the  centre 
C,  with  a  radius  equal  to  cr  V  or  a'  c'.  And  the  intersections  of  the  two 
oblique  faces  D'E'  and  F'  G'  will  obviously  be  each  projected  in  two  arcs 
of  an  ellipse  whose  major  axis  dg  is  equal  to  the  diameter  of  the  circle  a  c  o, 
and  the  minor  axis  is  the  vertical  projection  of  that  diameter,  as  represented 
at  e'  f  (fig.  2).  But  as  it  is  necessary  to  draw  small  portions  only  of  these 
curves,  the  following  method  may  be  employed. 
7 


98  GEOMETRICAL   PROJECTION. 

Draw  D  G ;  through  the  points  E,  F,  divide  the  portions  E  F  and  F  G 
respectively  into  the  same  number  of  equal  parts,  and  drawing  perpen- 
diculars through  the  points  of  division,  set  off  from  F  G  the  distances  from 
the  corresponding  points  in  E  F  to  the  circular  arc  E  C  F,  as  points  in  the 
elliptical  arc  required.  The  remaining  elliptical  arcs  should  be  traced  by 
the  same  method. 

Figs.  3  and  4. — Required  to  draw  the  lines  of  penetration  of  a  cylinder 
and  a  sphere,  the  centre  of  the  sphere  leing  without  the  axis  of  the  cylinder. 

Let  the  circle  D'  E'  I/  be  the  projection  of  the  base  of  the  given  cylin- 
der, the  elevation  of  which  is  shown  at  fig.  3,  and  let  A  B  be  the  diameter 
of  the  given  sphere.  If  a  plane,  as  c'  d',  be  drawn  parallel  to  the  vertical 
plane,  it  will  evidently  cut  the  cylinder  in  two  straight  lines  G  G',  H  IT, 
parallel  to  the  axis,  and  projected  vertically  from  the  points  G'  and  IF. 
This  plane  will  also  cut  the  sphere  in  a  circle  whose  diameter  is  equal  to 
c'  d',  and  which  is  to  be  described  from  the  centre  C  with  a  radius  of  half 
that  line  ;  its  intersection  with  the  lines  G  G'  and  H  H'  will  give  so  many 
points  in  the  curves  sought,  viz.,  G,  H,  I,  K. 

The  planes  a'  V  and  e'  f ,  which  are  tangents  to  the  cylinder,  furnish 
only  two  points  respectively  in  the  curves  ;  of  these  points  E  and  F  alone 
are  visible,  the  other  two,  L  and  M,  being  concealed  by  the  solid  ;  there- 
fore, the  planes  drawn  for  the  construction  of  the  curves  must  be  all  taken 
between  a'  V  and  e'f.  The  plane  which  passes  through  the  axis  of  the 
cylinder  cuts  the  sphere  in  a  circle  whose  projection  upon  the  vertical 
plane  will  meet  at  the  points  D,  N,  and  g,  h,  the  outlines  of  the  cylinder, 
to  which  the  curves  of  penetration  are  tangents. 

Figs.  5  and  6. — To  find  the  lines  of  penetration  of  a  truncated  cone 
and  a  prism. 

The .  straight  line  C  D  is  the  axis  of  a  truncated  cone,  which  is  repre- 
sented in  the  plan  by  two  circles  described  from  the  centre  C' ;  and  the 
horizontal  lines  M  N  and  M'  N'  are  the  projections  of  the  axis  of  a  prism 
of  which  the  base  is  square,  and  the  faces  respectively  parallel  and  per- 
pendicular to  the  planes  of  projection. 

In  laying  down  the  plan  of  this  solid,  it  is  supposed  to  be  inverted,  in 
order  that  the  smaller  end  of  the  cone,  and  the  lines  of  intersection  of  the 
lower  surface  F  G  of  the  prism  may  be  exhibited.  According  to  this 
arrangement,  the  letters  A'  and  B'  (fig.  6)  ought,  strictly  speaking,  to  be 
marked  at  the  points  I'  and  H/,  and  conversely  ;  but  as  it  is  quite  obvious 
that  the  part  above  M'  J\f  is  exactly  symmetrical  with  that  below  it,  the 
distribution  of  the  letters  of  reference  adopted  in  our  figures  can  lead  to  no 
confusion. 


GEOMETRICAL   PROJECTION.  99 

The  intersection  of  the  plane  F  G  with  the  cone  is  projected  horizon- 
tally in  a  circle  described  from  the  centre  C',  with  the  diameter  F'  G'. 
The  arcs  I'  F'  A'  and  H'  G'  B'  are  the  only  parts  of  this  circle  which  re- 
quire to  be  drawn. 

Figs.  7  and  8. — To  describe  the  curves  formed  ly  the  intersection  of  a 
cylinder  with  the  frustum  of  a  cone,  the  axes  of  the  two  solids  cutting  each 
Bother  at  right  angles. 

The  axes  of  the  solids  and  their  projections  are  laid  down  in  the  fignres 
precisely  as  in  the  preceding  example.  The  intersections  of  the  outlines 
of  the  cone  in  the  elevation  with  those  of  the  cylinder,  furnish,  obviously, 
four  points  in  the  curves  of  penetration ;  these  points  are  all  projected 
horizontally  upon  the  line  A'  B'.  Now,  suppose  a  plane,  as  a  o  (fig.  7), 
to  pass  horizontally  through  both  solids ;  its  intersection  with  the  cone 
will  be  a  circle  of  the  diameter  c  d,  while  the  cylinder  will  be  cut  in  two 
parallel  straight  lines,  represented  in  the  elevation  by  a  5,  and  whose  hori- 
zontal projection  may  be  determined  in  the  following  manner  : — Conceive 
a  vertical  plane  f  g,  cutting  the  cylinder  at  right  angles  to  its  axis,  and  let 
the  circle  g  ef  thereby  formed  be  described  from  the  intersection  of  the 
axes  of  the  two  solids  ;  the  line^'  h  will  now  represent,  in  this  position  of 
the  section,  the  distance  of  one  of  the  lines  sought  from  the  axis  of  the 
cylinder.  Now  set  off  this  distance  on  both  sides  of  the  point  A',  and 
through  the  points  Jc  and  a'  thus  obtained,  draw  straight  lines  parallel  to 
A'  B' ;  the  intersections  of  these  lines  with  the  circle  drawn  from  the 
centre  C'  of  the  diameter  c  d  will  give  four  points  m' ',  p',  n,  and  0,  which 
being  projected  vertically  upon  a  5,  determine  two  points  in  and^>  in  the 
curves  required. 

In  order  to  obtain  the  vertices  or  adjacent  limiting  points  of  the  curves, 
draw  from  the  vertex  of  the  cone  a  straight  line  t  e,  touching  the  circle  g  ef, 
and  let  a  horizontal  plane  be  supposed  to  pass  through  the  point  of  con- 
tact e.  Proceed  according  to  the  method  given  above  to  determine  the 
intersections  of  this  plane  with  each  of  the  solids  in  question,  the  four 
points  *',  /,  £,  and  s,  which  being  projected  vertically  upon  the  line  e  r, 
determine  the  vertices  i  and  r  required. 

OF    THE    HELIX. 

Plate  XII. — The  Helix  is  the  curve  described  upon  the  surface  of  a 
cylinder  by  a  point  revolving  round  it,  and  at  the  same  time  moving 
parallel  to  its  axis  by  a  certain  invariable  distance  during  each  revolution. 
This  distance  is  called  the  pitch  of  the  screw. 


100  GEOMETEICAL   PROJECTION. 

Figs.  1  and  2. — Required  to  construct  the  helical  curve  described  by  the 
point  A  upon  a  cylinder  projected  horizontally  in  the  circle  A!  C'  F;,  the 
pitch  being  represented  by  the  line  A.'  A3. 

Divide  the  pitch  A'  A3  into  any  number  of  equal  parts,  say  eight ;  and 
through  each  point  of  division,  1,  2,  3,  &c.,  draw  straight  lines  parallel  to 
the  ground  line.  Then  divide  the  circumference  A'  C'  F'  into  the  same 
number  of  parts ;  the  points  of  division  B',  C',  E',  F',  &c.,  will  be  the 
horizontal  projections  of  the  different  positions  of  the  given  point  during 
its  motion  round  the  cylinder.  Thus,  when  the  point  is  at  B'  in  the  plan, 
its  vertical  projection  will  be  the  point  of  intersection  B  of  the  perpendicu- 
lar drawn  through  B'  and  the  horizontal  drawn  through  the  first  point  of 
division.  Also,  when  the  point  arrives  at  C'  in  the  plan,  its  vertical  pro- 
jection is  the  point  C,  where  the  perpendicular  drawn  from  C'  cuts  the 
horizontal  passing  through  the  second  point  of  division,  and  so  on  for  all 
the  remaining  points.  The  curve  A  B  C  F  A3  drawn  through  all  the 
points  thus  obtained,  is  the  helix  required. 

Figs.  1  and  2. — To  draw  the  vertical  elevation  of  the  solid  contained 
between  two  helical  surfaces  and  two  concentric  cylinders. 

A  helical  surface  is  generated  by  the  revolution  of  a  straight  line 
round  the  axis  of  a  cylinder ;  its  outer  end  moving  in  a  helix,  and  the  line 
itself  forming  with  the  axis  a  constant  and  invariable  angle. 

Let  A'  C'  F7  and  K'  M'  O'  represent  the  concentric  bases  of  the  cylin- 
ders, whose  common  axis  S  T  is  vertical ;  the  curve  of  the  exterior  helix 
A  C  F  A3  is  the  first  to  be  drawn  according  to  tHe  method  above  shown. 
Then  having  set  off  from  A  to  A2  the  thickness  of  the  required  solid,  draw 
through  A2  another  helix  equal  and  similar  to  the  former.  Now  construct, 
as  above,  another  helix,  K  C  O,  of  the  same  pitch  as  the  last,  but  on  the- 
interior  icylinder ;  as  also  another,  K2  C2  O2,  equal  and  parallel  to  the 
former.  The  lines  A'  K',  B'  I/,  C'  M',  &c.,  represent  the  horizontal  pro- 
j  ections  of  the  various  positions  of  the  generating  straight  line,  which,  in 
the  present  example,  has  been  supposed  to  be  horizontal ;  and  these  lines 
are  projected  vertically  at  A  K,  B  L,  &c. 

It  will  be  observed^  that  in  the  position  A  K  the  generating  line  is 
projected  in  its  actual  length,  and  that  at  the  position  C'  M'  its  vertical 
projection  is  the  point  C.  The  same  remark  applies  to  the  generatrix  of 
the  second  helix.  The  parts  of  both  curves  which  are  visible  in  the  ele- 
vation may  be  easily  determined  by  inspection. 

Figs.  3  and  4. — To  determine  the  vertical  projection  of  the  solid  formed 
by  a  sphere  moving  in  a  helical  curve. 

Let  A'  C'  E'  be  the  base  of  a  cylinder,  upon  which  the  centre  point  C' 


GEOMETRICAL   PROJECTION.  101 

of  a  sphere  whose  radius  is  a!  C'  describes  a  helix,  which  is  projected  on 
the  vertical  plane  in  the  curve  A  C  E  J.  After  determining  as  above  the 

various  points  A,  B,  C,  D ,  in  this  curve,  draw  from  each  of  these 

points  as  centres,  circles  with  the  radius  a!  C' ;  the  circumferences  of  these 
circles  will  denote  the  various  positions  of  the  sphere  during  its  motion 
round  the  cylinder;  and  if  lines  be  drawn  touching  these  circles,  the 
curves  thereby  formed  will  constitute  the  figure  required.  One  of  these 
curves  will  disappear  at  O,  which  is  its  point  of  contact  with  the  circle 
described  from  the  point  E,  the  intersection  of  the  helix  with  the  perpen- 
dicular E  E' ;  it  will  again  reappear  at  the  point  I  when  it  becomes  a  tan- 
gent to  the  circle  described  from  the  point  J  in  the  prolongation  of  the 
line  A  A7.  The  exterior  and  interior  circles  (fig.  4)  represent  the  horizon- 
tal projection  of  the  solid  in  question. 

The  conical  helix  differs  from  the  cylindrical  one  in  that  it  is  described 
on  the  surface  of  a  cone  instead  of  on  that  of  a  cylinder  ;  but  the  construc- 
tion differs  but  slightly  from  the  one  described.  By  following  out  the  same 
principles,  helices  may  be  represented  as  lying  upon  spheres  or  any  other 
surfaces  of  revolution. 

In  the  arts  are  to  be  found  numerous  practical  applications  of  the  heli- 
cal curve,  as  wood  and  machine  screws,  geers,  and  staircases,  the  construc- 
tion of  which  will  be  still  farther  explained  under  their  appropriate  heads. 


102  THE  DEVELOPMENT  OF  SUEFACES. 


THE  DEVELOPMENT  OF  SURFACES. 

THE  development  of  the  surface  of  a  solid  is  the  drawing  or  unrolling 
on  a  plane  the  form  of  its  covering ;  and  if  that  form  be  cut  out  of  paper, 
it  would  exactly  fit  and  cover  the  surface  of  the  solid.  Frequently  in 
practice,  the  form  of  the  surface  of  a  solid  is  found  by  applying  paper,  or 
thin  sheet  brass  directly  to  the  solid,  and  cutting  it  to  fit.  Tin  and  copper- 
smiths, boiler-makers,  &c.,  are  continually  required  to  form  from  sheet 
metal  forms  analogous  to  solids  ;  to  execute  which  they  should  be  able  to 
construct  geometrically  the  development  of  the  surface  of  which  they  are 
to  make  the  form. 

The  development  of  the  surface  of  a  plain  cylinder  is  evidently  but  a 
plane  sheet,  of  which  the  circumference  is  one  dimension  whilst  its  length 
is  the  other. 


THE   DEVELOPMENT   OF   THE   STIRFACE   OF  INTERSECTED   CYLINDEES. 

Plate  XIIL,  figs.  1  and  2. — To  draw  the  surface  of  a  cylinder  formed 
~by  the  intersection  of  another  equal  cylinder,  as  the  knee  of  a  stove  pipe. 

Let  A  B  C  D  be  the  elevation  of  the  pipe  or  cylinder.  Above  A  B 
describe  the  semicircle  A'  4'  B'  of  the  same  diameter  as  the  pipe ;  divide  this 
semicircle  into  any  number  of  equal  parts,  eight  for  instance ;  through  these 
points  I7,  2',  3',  &c.,  draw  lines  parallel  to  side  A  C  of  pipe,  and  cutting 
the  line  C  D  of  the  intersection  of  the  two  cylinders.  Lay  off  A!'  B"  equal  to 
the  semicircle  A'  4'  B',  and  divided  into  the  same  number  of  equal  parts ; 
through  these  points  of  division  erect  perpendiculars  to  A"  B",  and  on  these 
perpendiculars  lay  off  the  distances  A"  C",  V  I",  2"  2",  3"  3"-,  and  so  on, 
corresponding  to  A  C,  1 1,  2  2,  3  3,  &c.,  in  preceding  figure.  Through  the 

points  C",  1",  2", D",  then  draw  connecting  lines,  and  we  have  the 

developed  surface  required.  It  is  to  be  remarked,  that  this  gives  but  one 
half  of  the  surface  of  the  pipe,  the  other  being  exactly  similar  to  it. 


THE  DEVELOPMENT  OF  SURFACES.  103 

Figs.  3  and  4. — To  develop,  the  surface  of  a  cylinder  intersected  by  an- 
other cylinder,  as  in  the  formation  of  a  "[pipe. 

The  construction  is  similar  to  the  preceding,  and  as  the  same  letters 
and  figures  are  preserved  relatively,  the  demonstration  will  be  easily 
understood  from  the  foregoing. 

The  development  of  the  surface  of  a  right  cone  (figs.  5  and  6).  From  C' 
(fig.  6)  as  a  centre,  with  a  radius  C'  A'  equal  to  the  inclined  side  A  C  of 
the  cone  (fig.  5),  describe  an  arc  of  a  circle  A'  B'  A"  ;  on  this  arc  lay  off 
the  distance  A'  B'  A"  equal  to  the  circumference  of  the  base  of  the  cone  ; 
connect  A'  C'  and  C'  A*,  and  A!  B'  A"  C'  is  the  developed  surface  re- 
quired. 

To  develop  the  surface  of  afrustrum  of  a  cone,  D  A  B  E  (fig.  5). 

On  fig.  6  develop  the  cut-off  cone  C  D  E  as  in  preceding  construction, 
and  we  have  A'  B'  A"  D"  E7  D  as  the  developed  surface  of  the  right 
frustrum. 

To  develop  the  surfqce  of  a  frustrum  of  a  cone,  when  the  cutting  plane 
a  1)  (fig.  5)  is  inclined  to  the  hase. 

On  A  B  the  base  describe  the  semicircle  A  3'  B ;  divide  the  semicircle 
into  any  number  of  equal  parts,  six  for  instance  ;  from  each  point  of  divi- 
sion 1',  2',  3',  4',  5',  let  fall  perpendiculars  to  the  base  ;  at  1,  2,  3,  4,  5,  con- 
nect each  of  these  last  points  with  the  apex  C.  Divide  now  the  arc  A'  B' 
(fig.  6)  into  six  equal  parts,  or  the  arc  A'  B'  A"  into  twelve ;  each  of  these 
parts  by  the  construction  is  equal  to  the  arc  A  1',  V  2'  (fig.  5) ;  connect 
these  points  of  division  with  the  point  C' ;  on  C'  A'  (fig.  6)  take  C'  a'  equal 
to  C  a  of  fig.  5,  a  being  the  point  at  which  the  plane  cuts  the  inclined  side 
of  the  cone  ;  in  the  same  way  on  C'  B',  lay  off  C'  V  equal  to  0  h. 

It  is  evident  that  all  the  lines  connecting  the  apex  C  with  the  base, 
included  within  the  two  inclined  sides,  are  represented  as  less  than  their 
actual  length  in  fig.  5,  and  must  be  projected  on  the  inclined  sides  to  de- 
termine their  absolute  dimensions;  project,  therefore,  the  points  1",  2//,  3", 
4",  5",  at  which  the  cutting  plane  intersects  the  lines  C  1,  C  2,  C  3,  -C  4, 
C  5,  by  drawing  parallels  to  the  base  through  these  points  to  the  inclined 
side  C  B'.  On  fig.  6  lay  off  C'  V" ',  0'  2"",  &c.,  equal  to  C  V",  C  2,"',  &e. 

(fig.  5) ;  connect  the  points  a',  \"" ,  2"", V, a",  and  we  have  the 

developed  surface  a'  A!  B'  A"  a"  V  required. 

To  develop  the  surface  of  a  sphere  or  hall  (figs.  189,  190). 

It  is  evident  that  the  surface  cannot  be  accurately  represented  on  a 
plane  surface.  It  is  done  approximately  by  a  number  of  gores.  Let  CAB 
(fig.  189)  be  the  eighth  of  a  hemisphere ;  on  C  D  describe  the  quarter 
circle  D  A  c  /  divide  the  arc  into  any  number  of  equal  parts,  six  for  in- 


104: 


THE  DEVELOPMENT   OF   SURFACES. 


stance ;  from  the  points  of  division  1,  2,  3, ...  let  fall  perpendiculars  on 
C  D,  and  from  the  intersections  with  this  line  describe  arcs  V 1",  2'  2",  3'  3", 

cutting  the  line  C  B  at  1",  2",  3", ;  on  the  straight  line  C'  D'  (fig. 

190),  lay  off  C'  D'  equal  to  the  arc  D  A  c,  with  as  many  equal  divi- 


Fig.  190. 


Fig.  1S9. 


sions  ;  then  from  either  side  of  this  line  lay  off  1'"  1"",  2'"  2""  : . . .  D'  B' 

equal  to  the  arcs  1'  1",  2'  2", D  B  (fig.  189).     Connect  the  points  C', 

1"",  2"", and  C'  A'  B'  is  the  developed  surface. 

It  is  to  "be  remarked,  that  in  the  preceding  demonstrations,  the  forms  are 
described  to  cover  the  surface  only ;  in  construc- 
tion, allowance  is  to  be  made  for  lap  by  the  addi- 
tion of  margins  on  each  side  as  necessary.  It 
is  found  difficult  in  the  formation  of  hemispherical 
ends  of  boilers,  to  bring  all  the  gores  together  at 
the  apex ;  it  is  usual,  therefore,  to  make  them,  as 
shown  (fig.  191),  by  cutting  short  the  gores,  and 
FI«.  i9L  surmounting  the  centre  with  cap  piece. 


MECHANICS. 


105 


MECHANICS. 

THE  profession  of  an  architectural  or  mechanical  draughtsman  should 
embrace  not  merely  the  mere  copying  of  examples  which  may  be  furnished 
him,  but  also  the  designing  of  new  edifices  and  machines,  in  which  he  may 
draw  from  the  results  of  his  own  experience ;  from  good  models,  by  col- 
lating suitable  parts  from  divers  designs ;  or  by  the  rules  of  mechanics, 
proportioning'  the  parts  according  to  the  magnitude  and  direction  of  the 
strains  to  which  they  are  to  be  subject,  arid  the  materials  of  which  they  are 
to  be  composed;  introducing  as  much  of  ornament  as  the  subject  may 
require. 


THE   MECHANICAL   POWEES. 

The  simple  machines  or  mechanical 
powers  which  enter  as  elements  into  the 
composition  of  all  machinery  are  the 
lever,  the  pulley r,  and  the  inclined  plane, 
to  which  some  add  the  toggle-joint  and 
the  hydrostatic  press.  The  lever  em- 
braces the  wheel  and  axle,  and  the  in- 
clined plane  the  wedge  and  screw. 

It  is  usual  to  regard  the  lever  as  of 
three  kinds,  distinguished  by  the  relative 
position  of  power  P,  weight  W,  and  ful- 
crum F.  In  the  first  the  fulcrum  is  be- 
tween the  power  and  the  weight ;  in  the 
2d,  the  weight  is  between  the  power  and 
the  fulcrum ;  and  in  the  3d,  the  power  is 
between  the  weight  and  the  fulcrum. 
This  division  is  rather  nominal  than  real; 
all  applications  of  the  lever  may  be  re- 
solved under  one  general  rule :  that  the 


IST  CLASS. 


W    F 


CLASS. 


Q 


106  MECHANICS. 

intermediate  weight,  pressure,  or  tension,  is  equal  to  the  sum  of  the  outside 
ones,  and  the  comparative  size  of  these  last  two  depends  on  their  position 
in  reference  to  the  first.  Let  the  intermediate  pressure  be  exerted  through 

the  means  of  a  spring  balance  c,  Fig. 
192,  it  will  mark  the  sum  of  the 
weights  a  and  ft. 

x _,  Calling  x  the  distance  from  a  to 

\l\  |_fj      C,  and  y  from  c  to  ft/  knowing  any 

FIG.  192.  three  of  the  four,  #,  ft,  x  and  y,  the 

fourth  may  be  found  ;  for  a  multiplied  by  x  is  equal  to  ft  multiplied  by  y 
(ax=ly).  Thus,  if  a  be  6  Ibs.,  x  5  ft.,  and  y  10  ft.,  then  6  multiplied  by 
5=30,  is  equal  to  10  multiplied  by  ft/  therefore  f£,or  3  Ibs.,  is  equal  to  ft, 
and  the  load  at  c  is  6-f  3=9  Ibs. 

As  c  divides  the  lever  proportionally  to  the  weights,  and  as  it  is  the 
sum  of  the  weights,  knowing  c,  x,  and  y,  a  and  ft  may  be  calculated.  Thus, 
if  the  load  at  c  be  100  Ibs.,  and  x  7  ft.,  and  y  3  ft.,  then  one  of  the  weights 
will  be  T7o  of  c,  or  70  Ibs.,  and  the  other  T3o,  or  30  Ibs, ;  and,  as  the  heavier 
weight  is  at  the  shortest  end  of  the  lever,  ft  will  be  70  Ibs.,  and  a  30  Ibs. 
a  is  to  ft  as  3  to  7,  but  x  is  to  y  as  7  to  3 ;  therefore,  the  weights  a  and  ft 
are  to  each  other  as  the  opposite  ends  of  the  lever,  as  ytox;  or  the  point 
c  divides  the  lever  in  the  inverse  proportion  of  the  weights. 

If  either  a  or  ft  be  known,  and  the  lengths  of  the  lever  x  and  y,  the 
load  at  c  may  be  calculated  directly,  c  is  equal  to  either  weight,  a  or  ft, 
multiplied  by  the  whole  length  of  the  lever,  divided  by  y  or  a?,  the  part  of 

the  lever  opposite  the  known  weight,  c  =  —  —    +     .    It  will  be  readily 

understood  that  a,  ft,  or  c  may  be  considered  at  option,  either  fulcrum, 
weight,  or  power,  according  to  the  requirements  of  the  mechanism. 

The  Wheel  and  Axle. — If  a  weight,  P,  be  suspended  from  the  periphery 
of  a  wheel,  fig.  193,  whilst  another  weight,  "W,  is  suspended  on  the  oppo- 
site side  of  a  barrel  or  axle  attached  to  the  wheel,  the  principle  of  action  is 
the  same  as  that  of  the  lever.  .  P  multiplied  by  its  length  of  lever  or  radius 
ca  of  the  wheel  is  equal  to  "W  multiplied  by  its  length  of  lever  or  radius 
of  the  axle  cb  /  the  axis  c  is  the  fulcrum.  If  a  movement  downward  be 
communicated  to  P,  as  shown  by  the  dotted  line,  a  rotary  motion  is  given 
to  the  wheel  and  axle ;  the  cord  of  P  is  unwound  whilst  that  of  W  is 
wound  up,  but  P  is  still  suspended  from  a  and  W  from  ft  /  the  leverage,  or 
distance  from  the  fulcrum,  of  each  is  the  same  as  at  first.  The  wheel  and 
axle  is  a  lever  of  continuous  and  uniform  action.  Since  the  wheel  has  a 
larger  circumference  than  the  axle,  by  their  revolution  more  cord  will  be 


MECHANICS. 


107 


unwound  from  the  former  than  is  wound  up  on  the  latter,  P  will  descend 
faster  thanW  is  raised,  in  the  proportion  of  the  circumference  of  the 

wheel  to  that  of  the  axle,  or  of  their  radii  ca  to 

,.«-.  ••**""          "**«< 

cl.     When  P  has  reached  the  position  if]  W 


Fio.  193. 


will  have  reached  I  W  \ .    If  c  a  be  4  times  c  5,  then  / 
P  will  have  moved  4  times  the  distance  that  ~W   \ 
has.     The  movement  is  directly  as  the  length  of     " 
the  levers,  or  the  radii  of  the  points  of  suspension. 
It  will  be  perceived,  therefore,  to  move  a  large 
weight  by  the  means  of  a  smaller  one,  that  the 
smaller  must  move  through  the  most  space,  and 
that  the  spaces  described  are  as  the  opposite  ends 
of  the  lever,  or  inversely  as  the  weights. 

It  is  the  fundamental  principle  of  the  action  of 
all  mechanical  powers,  that  whatever  is  "  gained 
in  power,"  as  it  is  said,  is  lost  in  space  travelled ; 
that,  if  a  weight  is  to  be  raised  a  certain  number 
of  feet,  the  force  exerted  to  do  this  must  always 
be  equal  to  the  product  of  the  weight  by  the 
height  to  which  it  is  to  be  raised ;  thus,  if  200  Ibs.  are  to  be  raised  50  ft., 
the  force  exerted  to  do  this  must  be  equal  to  a  weight,  which,  if  multiplied 
by  its  fall,  will  be  equal  to  the  product  200  x  50,  or  10,000 ;  and  it  is 
immaterial  whether  the  force  be  a  weight  of  10,000  Ibs.  falling  one  ft.,  or 
1  Ib.  10,000  ft. 

It  is  now  common  to  refer  all  forces  exerted  to  a  unit  of  Ibs.  ft. ;  that 
is,  1  Ib.  falling  1  ft. ;  and  the  effect  to  the  same  unit  of  Ibs.  ft;,  1  Ib. 
raised  1  ft.  Thus,  in  the  example  above,  the  force  exerted  or  power  is 
10,000  Ibs.  ft.  falling ;  the  effect  10,000  Ibs.  ft.  raised.  In  practice,  the 
Ibs.  ft.  of  force  exerted  must  always  be  more  than  the  Ibs.  ft.  of  effect  pro- 
duced ;  that  is,  there  must  be  some  excess  of  the  former  to  produce  move- 
ment, and  to  overcome  resistance  and  friction  of  parts. 

The  measure  of  any  force,  as  represented  by  falling  weight,  is  termed 
the  absolute  power  of  that  force ;  the  resulting  force,  or  useful  effect  for  the 
purposes  for  which  it  is  applied,  is  called  the  effective  power. 

The  Pulley. — The  single  fixed  pulley,  fig.  194,  consists  of  a  single 
grooved  wheel  movable  on  a  pin  or  axis ;  called  fixed,  because  the  strap, 
through  which  the  pin  passes,  is  attached  to  some  fixed  object.  A  rope 
passes  over  the  wheel  in  the  groove ;  on  one  side  the  force  is  exerted,  and 
on  the  other  the  weight  is  attached  and  raised.  It  may  be  considered  a 


108 


wheel  and  axle  of  equal  diameters,  or  as  a  lever  in 
which  the  two  sides  are  equal,  the  pin  being  the  ful- 
crum. P,  the  force  exerted,  must  therefore  be  equal  to 
the  weight,  "W,  raised ;  and,  if  movement  takes  place, 
"W  will  rise  as  much  as  P  descends. 

The  fixed  pulley  is  used  for  its  convenience  in  the 
application  of  the  force ;  it  may  be  easier  to  pull  down 
than  up,  for  instance ;  but  the  Ibs.  of  force  must  be 
FIO.  iw.  equal  to  the  lbs<  of  effect     The  tension  on  the  rope  is 

equal  to  either  the  force  or  weight. 

Fig.  195  is  a  combination  of  a  fixed  pulley  A,  and  movable  pulley  B. 
The  simplest  way  to  arrive  at  the  principle  of  this  combination  is  to  con- 
sider its  action.  Let  P  be  pulled  down,  say  2  feet ; 
the  length  of  rope  drawn  to  this  side  of  the  pulley 
must  be  furnished  from  the  opposite  side.  On  that 
side  there  is  a  loop,  in  which  the  movable  pulley 
with  the  weight  W  attached  is  suspended.  Each 
side  of  this  loop,  2  and  3,  must  go  to  make  up  the 
2  ft.  for  the  side  or  end  1.  Cords  2  and  3  will 
therefore  furnish  each  1  ft.  As  these  cords  are 
shortened  1  ft.  the  weight  W  is  raised  1  ft.,  and  as 
the  movement  of  "W  is  but  1  ft.  for  the  2  ft.  of  P,  W  must  be  twice  that  of 
P,  because  the  2  Ibs.  ft.  of  P  must  equal  Ibs.  ft.  of  W. 

In  the  combination  of  pulleys,  Fig.  196.  Let  P  be  pulled,  say  3  ft., 
,,/,.','.-///».  then  this  length  of  rope  drawn  from  the  opposite 
side  of  the  pulley  is  distributed  over  the  3  cords  2, 
.  3,  4,  and  the  weight  W  is  raised  1  ft. ;  conse- 
quently, the  weight  W  is  3  times  that  of  P.  The 
cord  1  supports  P,  the  cords  2,  3,  4,  the  weight  "W, 
or  3  times  P ;  consequently,  the  tension  on  every 
cord  is  alike.  The  same  rope  passing  freely  round 
pulleys  must  have  the  same  tension  throughout ;  so 
that,  to  determine  the  relation  of  W  to  P,  count  the 
number  of  cords  which  sustain  the  weight.  Thus 
in  Fig.  197  the  weight  is  sustained  by  4  cords ;  con- 
sequently it  is  4  times  the  tension  of  the  cord,  or  4 
times  the  force  P.  In  order  not  to  confuse  the 
cords,  the  pulleys  are  represented  as  in  the  figures ;  but,  in  construction, 
the  pulleys,  or  sheaves,  are  usually  of  the  same  diameter,  and  those  in  con- 
nection, as  A  and  B,  and  C  and  D,  run  on  the  same  pin. 


FIG.  195. 


FIO.  196. 


MECHANICS. 


109 


The  Inclined  Plane. — To  support  a  weight  by  means  of  a  single  fixed 
pulley,  the  force  must  be  equal  to  the  weight.  Suppose  the  weight, 
instead  of  hanging  freely,  to  rest  upon  an  inclined  plane  5  d,  Fig.  198 ;  if 
motion  ensue,  to  raise  the  weight  "W  the  height  a  5,  the  rope  transferred 
from  the  weight  side  of  the  pulley  will  be  equal  to  bd,  and  P  will  have 
consequently  fallen  this  amount;  thus,  if  Id  be  6  ft.,  and  al  1  ft,  whilst 
W  is  raised  1  ft.  P  has  descended  6  ft.,  and  as  Ibs.  ft.  of  power  must  equal 
Ibs.  ft.  of  effect,  P  will  be  £  of  W ;  and,  by  reference  to  the  figure,  P  is  to 
W  as  a 5  is  to  fid,  or  as  the  height  of  the  incline  is  to  its  length.  If  the 


FIG.  198.  FIG.  199. 

end  of  the  plane  d  be  raised,  till  it  becomes  horizontal,  the  whole  weight 
would  rest  on  the  plane,  and  no  force  would  be  necessary  at  P  to  keep  it 
in  position ;  if  the  plane  be  revolved  on  5,  till  it  becomes  perpendicular, 
then  the  weight  is  not  supported  by  the  plane  at  all,  but  it  is  wholly 
dependent  on  the  force  P,  and  is  equal  to  it.  Between  the  limits,  there- 
fore, of  a  level  and  a  perpendicular  plane,  to  support  a  given  weight  W, 
the  force  P  varies  from  nothing  to  an  equality  with  the  weight. 

The  construction,  Fig.  199,  illustrates  the  principle  of  the  wedge,  which 
is  but  a  movable  inclined  plane ;  if  the  wedge  be  drawn  forward  by  the 
weight  P,  and  the  weight  W  be  kept  from 
sliding  laterally,  the  fall  of  P,  a  distance 
equal  to  a  d,  will  raise  the  weight  W  a 
height  eft.  P  will  therefore  be  to  W,  as  cb 
is  to  a  d.  For  example,  if  the  length  of  the 
wedge  ad  be  10  ft.,  and  the  back  cb  2  feet, 
then  P  will  be  to  W,  as  2  to  10,  or  }  of  it. 

Let  the  inclined  plane  aid,  Fig.  198,  be 
bent  round,  and  attached  to  the  drum  A, 
Fig.  200,  to  which  motion  of  revolution  on  FIO>  200. 

its  axis  is  given,  by  the  unwinding  of  the  turns  of  a  cord  from  around  its 
periphery,  through  the  action  of  a  weight  P  suspended  from  a  cord  passing 
over  a  pulley.  If  the  weight  W  be  retained  in  its  vertical  position,  by  the 
revolution  of  the  drum  it  will  be  forced  up  the  incline,  and  when  the  cord 
has  unwound  one-half  turn  from  the  drum,  and  consequently  the  weight  P 


110 


MECHANICS. 


descended  a  distance  ce  equal  to  one-half  the  circumference  of  the  drum, 
the  weight  W  has  been  raised  to  the  height  a  I  by 
the  half  revolution  of  the  plane ;  P  must  therefore 
be  to  W  as  a  b  is  to  one-half  the  circumference. 
Extend  the  inclined  plane  so  as  to  encircle  the 
;pdrum,  Fig.  201.  The  figure  illustrates  the 
mechanism  of  the  screw,  which  may  be  considered 
as  formed  by  wrapping  a  fillet-band  or  thread 
around  a  cylinder  at  a  uniform  inclination  to  the  axis.  In  practice,  the 
screw  or  nut,  as  the  case  may  be,  is  moved  by  means  of  a  force  applied  at 
the  extremity  of  a  lever,  a  complete  revolution  raises  the  weight  the  dis- 
tance from  the  top  of  one  thread  to  the  top  of  the  one  above,  or  the  pitch. 
If  the  force  be  always  exerted  at  right  angles  to  the  lever,  Fig.  202,  the 
lever  may  be  considered  the  radius  of  a  wheel,  at  the 
circumference  of  which  the  force  is  applied.  Thus,  if 
the  lever  be  3  ft.  long,  the  diameter  of  the  circle 
would  be  6  ft.,  and  the  circumference  6  x  3.1416  or 
18  TVo  ft-,  if  the  pitch  be  1  inch,  or  TV  of  a  foot,  then 
the  force  would  be  to  the  weight  as  TV  is  to  18.85.; 
and  if  the  force  be  1  lb.,  the  weight  would  be 
226.20  Ibs. 

PARALLEL  FORCES. — If  two  horses  be  harnessed  to  a  load,  the  effect  is 
to  draw  a  load  equal  to  the  sum  of  their  forces  exerted  in  a  line  opposed 
to  the  resistance  of  the  load  at  the  point  where  the  whiffle-tree  is  attached 
to  the  load.  The  forces,  both  of  exertion  and  resistance,  act  in  parallel 
lines,  and  the  resultant  of  the  two  forces,  which  is  their  sum,  and  counter- 
balances the  third,  must  be  applied  at  a  point  intermediate  between, 
and  distant  from  each  of  them,  inversely  as  the  forces  exerted.  The 
composition  of  two  parallel  forces  acting  in  the  same  direction  is  to  be 
solved  as  an  example  of  a  lever,  with  an  intermediate  fulcrum.  The 
forces  are  represented  by  the  weights,  and  the  point  of  application  of  the 
resultant  by  the  fulcrum,  which  is  acted  on,  as  if  there  were  but  a  single 
force,  equal  to  the  sum  of  the  two,  opposed  to  it. 

The  resultant,  of  any  number  of  parallel  forces,  acting  in  one  direction, 
is  equal  to  their  sum,  acting  in  the  same  direction  at  some  intermediate 
point ;  that  is,  the  effect  of  all  these  forces  is  just  the  same,  as  if  there  were 
but  one  force,  equal  to  their  sum,  acting  at  tliis  point,  and  is  balanced  by 
an  equal  force  acting  in  the  opposite  direction.  This  central  point  may  be 
determined  by  finding  the  resultant,  i.  e.,  the  sum,  and  the  point  of  appli- 


MECHAOTCS. 


Ill 


cation  for  any  two  of  the  forces,  and  then  of  other  two,  the  resultants  thus 
determined  being  again  added  together  like  simple  forces. 

As  parallel  forces  can  be  added  together ;  reciprocally,  they  can  also  be 
divided.  The  single  force,  acting  intermediately,  may  be  resolved  into 
forces  acting  at  the  ends  of  a  lever,  whose  sum,  whatever  their  number, 
will  be  equal  to  the  central  force. 

Inclined  Forces. — When  two  men  of  equal  strength  pull  directly  opposite 
to  each  other,  the  resultant  is  nothing.     Let  a  third  take  hold  of  the  centre 
of  the  rope,  and  pull  at  right  angles  to 
the  rope;  he  will  make  an  angle  in  the 
rope  and  the  two  others  are  now  pulling  in 
directions  inclined  to  each  other.     The  less 
the  force  exerted  at  the  centre,  the  less  the 
flexure  in  the  rope ;  but  when  it  becomes 

equal  to  the  other  two,  the  two,  to  balance  it,  must  pull  directly  against  it, 
bringing  the  ends  of  the  rope  together,  and  acting  as  parallel  forces.  Be- 
tween the  smallest  force  and  the  largest,  that  can  be  exerted  at  the  centre 
and  maintain  a  balance  or  equilibrium,  the  ends  of  the  rope  assume  all 
varieties  of  angles,  which  angles  bear  definite  relations  to  the  forces. 

Kepresent  these  forces  by  weights,  as  in  Fig.  203  ;  let  P  and  "W  be  the 
extreme  forces,  acting  over  pulleys,  and  tending  to  draw  the  rope  straight, 
which  the  weight  C  prevents:  to 
find  what  must  be  the  weight  of  0 
to  balance  the  others.  Above  the 
rope  draw  the  lines  cp  and  cw, 
parallel  to  the  rope,  from  c  lay  off 
on  cp  parts  of  an  inch,  equal  to 
number  of  pounds  or  units  of 
weight  in  P;  that  is,  if  P  be  5 
Ibs.,  and  W  6  Ibs.,  lay  off  cp  f ,  T\, 
or  any  fractions  of  an  inch  that 
may  be  the  most  convenient,  say  f ;  and  c  w  £  of  an  inch ;  draw  ^? a  parallel 
to  cw,  and  wa  parallel  to  cp,  connect  ac  /  if  the  length  ac  be  measured 
in  Stlis  of  an  inch,  or  whatever  fractional  parts  may  be  adopted  for  the 
other  weights,  it  will  represent  the  weight  of  C  in  Ibs.  (in  this  case  about 
5£  parts,  or  Ibs.),  and  the  direction  in  which  C  acts ;  the  work  of  the 
weights  P  and  W  is  to  support  C,  and  this  would  be  done  by  a  force  equal 
to  C,  5|  Ibs.,  acting  in  a*  line  c  a  directly  above  it.  Therefore,  the  force 
opposed  to  the  direction  of  0,  and  equal  to  it,  will  be  the  resultant  of  the 
two  forces,  P  and  W  acting  at  an  angle  to  each  other. 


FIG.  203. 


112 


MECHANICS. 


As  two  forces  may 
be  compounded,  so  con- 
versely one  force  may  be 
resolved  into  two;  thus, 
let  the  weight  P,  Fig. 
204,  be  supported  by  two 
inclined  rafters  CA  and 
C  B.  Each  resists  a  part 
of  the  force  exerted  by 
the  weight  P.  To  find 
the  force  exerted  against 
the  abutments  A  and  B, 
in  the  direction  of  CA 
and  CB,  draw  cA.'  par- 
allel to  C  A,  cB' to  C  B, 
and  cd,  a  continuation  of 
the  line  C  P,  the  direction 

in  which  the  weight  P  acts ;  lay  off  c  d  from  a  scale  of  equal  parts,  a 
length  which  will  represent  the  number  of  Ibs.,  or  whatever  unit  of  weight 
there  may  be  in  the  weight  P;  draw  da  parallel  to  cB',  and  d b  parallel 
to  cA'j  ca,  measured  on  the  scale  of  equal  parts  adopted,  will  represent 
the  Ibs.  or  units  of  weight  exerted  against  A  in  the  direction  of  C  A,  and 
c  I  the  Ibs.  or  units  of  weight  exerted  against  B  in  the  direction  of  C  B. 

This  method  of  finding  the  resultant  of  two  forces,  or  the  components 
of  one  force,  is  called  the  parallelogram  of  forces.  If  two  sides  of  a  paral- 
lelogram represent  two  forces  in  magnitude  and  direction,  the  resultant  of 
these  two  forces  will  be  represented  in  magnitude  and  direction  by  the 
diagonal  of  the  parallelogram,  and  conversely. 

The  sum  of  ac  and  cb  is  greater  than  cd j  that  is,  the  weight  P  exerts 
a  greater  force  in  the  direction  of  the  lines  C  A  and  C  B,  against  A  and 
B,  than  its  own  weight ;  but  the  down  pressure  upon  A  and  B  is  only  equal 
to  the  weight  of  P  and  of  the  rafters  which  support  it,  which  last,  in  the 
present  consideration,  is  neglected.  Resolve  cb,  the  force  acting  on  B  in 
the  direction  of  c  B',  into  gl)  or  c  e  the  downward  pressure,  and  c  g  or  e  5 
the  horizontal  thrust  on  the  abutment  B,  and  c  a  into  cf  and  fa.  To 
decompose  a  force,  form  a  triangle  with  the  direction  of  the  other  forces, 
upon  the  line  representing  the  magnitude  and  direction  of  the  given  force. 
ce  represents  the  weight  on  B,  cforde  the  weight  on  A ;  cd,  or  ce  +  de, 
the  whole  weight  P ;  therefore,  the  weight  upon  the  two  abutments  A  and 
B  is  equal  to  the  whole  weight  of  P. 


MECHANICS. 


113 


Connect  A'  B',  and  extend  the  line  c  d  to  E.  c  e  is  to  e  d  as  A'  E  is  to 
E  B/  and  as  the  weight  is  distributed  on  the  abutments  A  and  B  in  propor- 
tion to  ed  and  ce,  it  will  be  also  as  B'E  is  to  A'E ;  or  the  whole  construc- 
tion may  be  considered  a  lever,  in  which  the  weight  is  suspended  at  E  and 
distributed  between  the  supports  inversely  as  its  distance  from  them,  af 
and  e  ft  represent  the  horizontal  thrust  on  the  abutments  A  and  B,  but  as 
af  and  b  e  are  equal,  the  force  tending  to  separate  them,  or  to  tear  asunder 
A'  B'  if  a  tie-rod,  would  be  represented  by  either  of  them. 

In  the  application  of  forces  to  a  lever  we  have  considered  them  parallel 
and  perpendicular  to  the  lever ;  if  they  are  inclined,  they  may  be  resolved 
into  forces  acting  perpendicular  to  and  in  the  direction  of  the  lever,  or 
they  may  simply  be  referred  to  the  fulcrum,  as  the  axis  of  a  wheel  and  axle, 


of  which  the  perpendiculars  let  fall  from  the  fulcrum  upon  the  direction  of 
the  forces  are  the  radii.  Thus,  if  the  levers,  Fig.  205,  be  acted  on  by 
forces  of  which  the  direction  is  shown  by  the  arrows,  the  leverage,  as  it  is 
called,  or  effective  length  of  arm  at  which  they  act,  are  the  perpendiculars 
Fa  and  F  J,  on  the  directions  of  the  forces. 

Centre  of  Gravity. — The  action  of  weight,  or  its  tendency 
downward  to  the  earth,  is  called  gravity.  Let  a  mass  P,  Fig. 
206,  be  suspended  by  a  cord,  each  particle  of  the  mass  is  acted 
upon  by  gravity,  like  an  innumerable  number  of  threads  pulling 
it  downward,  and  all  these  parallel  forces  may  be  resolved  into 
a  single  force  opposed  to  the  direction  of  the  string,  and  equal 
the  sum  of  all  the  forces  or  the  weight. 

Suspend  the  mass  C,  Fig.  207,  by  a  cord  from  P ;  the  line 


Fio.  206. 


FIG.  208. 


114 


MECHANICS. 


PM  will  represent  the  direction  of  the  resultant  through  the  mass.  Sus- 
pend the  same  mass  again  from  Q,  the  resultant  will  now  take  the  direc- 
tion Q  K,  and  the  two  resultants  have  one  point  C,  their  intersection  in 
common ;  this  point  is  called  the  centre  of  gravity ;  all  the  weight  may  be 
supposed  concentrated  at  this  point. 

A  body  placed  on  a  horizontal  plane  will  fall  over,  unless  the  vertical 
line  passing  through  its  centre  of  gravity  fall  within  its  base ;  thus,  in  Fig. 
208,  the  body  will  stand  firmly,  whilst  in  Fig.  209  it  will  fall  over.  Per- 
sons carrying  loads,  Fig.  210,  adjust  their  position  in- 
sensibly so  that  the  vertical  line  from  the  common  cen- 
tre of  gravity,  ^,  of  their  own  bodies  G,  and  of  their 
load  H,  should  fall  within  the  area  bounded  by  their  feet. 
The  body  is  thrown  to  the  side  opposite  the  load. 

When  bodies  are  of  symmetrical  forms  and  homo- 
geneous, the  determination  of  the  centre  of  gravity  is 
FIG.  210.  the  finding  the  centre  of  the  figure.    The  centre  of 

gravity  of  a  triangle  is  in  a  line  drawn  from  its  summit  to  the  middle  of 
the  opposite  side,  and  at  £  of  its  length  from  the  base. 

The  centre  of  gravity  of  any  quadrilateral  may  be  found  by  dividing  it 
into  two  triangles,  and  finding  the  centre  of  gravity  of  these  triangles,  con- 
necting them  by  a  straight  line,  and  dividing  this  line  into  parts  inversely 
proportional  to  the  surfaces  of  the  two  triangles,  the  point  of  division 
being  the  centre  of  gravity  of  the  figure.  In  the  same 
way  any  polygon  may  be  subdivided  into  triangles 
of  which  the  centres  of  gravity  may  be  found,  and  re- 
solved by  connections  with  each  other. 

The  centre  of  gravity  of  the  triangular  pyramid, 
Fig.  211,  is  in  the  straight  line  A  E,  connecting  the  apex 
A  with  the  centre  of  gravity  of  the  base  triangle  BCD, 
FIO.  211.  and  distant  £  of  the  length  of  the  line  A  E  from  E. 

The  centre  of  gravity  of  solids,  which  may  be  divided  into  symmetrical 
figures  and  pyramids,  as  for  all  practical  purposes  most  may  be,  can  be 
found  by  determining  the  centre  of  gravity  of  each  of  the  solids  of  which 
it  is  compounded,  and  then  compounding  them,  observing  that  each  centre 
of  gravity  represents  the  solid  contents" of  its  own  mass  or  masses  of  which 
it  may  be  composed.  The  centre  of  gravity  of  bodies  enclpsed  by  more  or 
less  regular  contours,  as  a  ship  for  instance,  is  determined  by  dividing  it 
into  parallel  and  equidistant  sections,  finding  the  centre  of  gravity  of  each, 
and  compounding  them  into  a  single  one. 


MECHANICS.  115 

FRICTION,    AND   THE   LIMITING   ANGLE   OF   RESISTANCE. 

Suppose  a  mass  A  (fig.  212)  be  pressed  upon  another,  B,  by  means  of 
a  force  acting  in  a  direction  perpendicular  to  the  common  surface  of  the 
two  bodies,  and  let  a  second  force  Q  act  also  upon  it  in  a  direction  paral- 
lel to  this  surface.  Then,  since  the  forces  B£and  Q  act  in  directions  per- 
pendicular to  each  other,  they  manifestly  cannot  counteract  one  another, 
and  it  would  be  expected  that  the  body  should  move  in  the  direction  of 
the  second  force.  This,  however,  is  not  always  the  case  ;  except  the  force 
Q  exceed  a  certain  limit,  no  motion  ensues.  Some  new  force  F,  therefore, 
has  been  produced  in  the  system,  counteracting  the  force  Q  ;  that  force  is 
called  friction.  It  acts  always  in  a  direction  parallel  to  the  surfaces  in 
contact,  and  is  always  for  surfaces  of  the  same  nature  the  same  fraction, 
or  part  of  the  force  P  by  which  these  are  pressed  together,  whatever  be  the 
amount  of  that  force,  or  whatever  the  extent  of  surfaces  in  contact.  This 
fraction  is  called  the  coefficient  of  friction.  Whilst  it  is  thus  the  same  for 
the  same  surfaces,  whatever- be  the  extent  of  the  surfaces,  or  the  force  with 
which  they  are  pressed  together,  it  is  different  for  different  surfaces. 

Construct  the  parallelogram  of  forces  P  P"  Q  M  (fig.  213).  P"  M  rep- 
resent the  resultant  of  the  two  forces  P  and  Q.  The  actual  friction  is 
always  a  certain  given  fraction  of  P  acting  parallel  to  the  impressed  sur- 
face. Take  M  Q'  equal  to  this  given  fraction  of  P  M,  complete  the  paral- 
lelogram, and  draw  the  diagonal  P'  M.  Since,  then,  M  Q'  represents  the 
friction  of  the  body  upon  the  plane,  or  the  force  called  into  action  by  P  M, 
which  opposes  the  motion  of  the  body ;  since,  moreover,  Q  M  represents 


^  F 


ft'  Q  C         M  A 

Fig.  212.  Fig.  213.  Fig.  214. 

the  force  tending  to  produce  motion,  it  follows  that  the  body  will  or  will 
not  move,  according  as  Q  M  is  greater  or  less  than  M  Q',  or  as  the  angle 
P  M  P"  is  greater  or  less  than  P'  M  P.  The  angle  P'  M  P  is  called  the 
limiting  angle  of  resistance.  It  depends  upon  the  coefficient  of  friction, 
and  is  therefore  the  same  for  the  same  surfaces,  whatever  be  the  actual 
amount  of  the  impressed  force  P. 

Hence  it  appears,  that  force  impressed  upon  the  surface  of  a  solid  body 
at  rest,  by  the  intervention  of  another  solid  body,  will  be  destroyed,  pro- 


116 


vided  the  angle  which  the  direction  of  that  force  makes  with  the  perpen- 
dicular to  the  surface  does  not  exceed  a  certain  angle  called  the  limiting 
angle  of  the  resistance  at  that  surface,  and  this  is  true,  however  great  the 
force  may  be.  Also,  that  if  the  direction  of  the  impressed  force  lie  with- 
out this  angle,  it  cannot  be  sustained  by  the  resistance  of  the  surfaces  in 
contact,  and  that  this  is  true,  however  small  the  force  may  be. 

Suppose  a  heavy  mass  (fig.  214),  whose  centre  of  gravity  is  G,  to  be 
placed  on  an  inclined  plane  A  B.  The  whole  pressure  of  the  mass  may 
be  supposed  to  act  in  the  direction  of  the  vertical  line  G  M,  and  this  pres- 
sure will  be  just  destroyed  by  the  resistance  of  the  surface  of  the  plane 
when  the  angle  G'  P  Q,  which  G  P  makes  with  the  perpendicular  P  Q,  is 
equal  to  the  limiting  angle  of  resistance.  A  mass  of  any  substance  will, 
therefore,  just  be  sustained  on  an  inclined  plane,  without  slipping,  when 
the  inclination  of  the  plane  is  equal  to  the  limiting  angle  of  the  resistance 
of  the  surfaces  in  contact ;  that  is,  when  the  angle  B  A  C  is  equal  to  the 
angle  G  P  Q. 

EXPERIMENTS    ON   FRICTION,   BY  M.  MORIN. 


WITHOUT  UNGUENTS. 

UNCTUOUS  SURFACES. 

PRICTION    OF   MO- 

FRICTION  OF   QUI- 

FRICTION  OP  MO- 

FRICTION   OF  QUI- 

SURFACES OF  CONTACT. 

TION. 

ESCENCE. 

TION. 

ESCENCE. 

Co-efflci- 

Limiting 

Co-effici- 

Limiting 

Co-effici- 

Limiting 

Co-effici- 

Limiting 

ent  of  fric- 

angle of 

ent  of  fric- 

angle of 

ent  of  fric- 

angle of 

ent  of  fric 

angle  of 

tion. 

resistance. 

tion. 

resistance. 

tion. 

resistance. 

tion. 

resistance. 

Oak  upon  oak,  fibres  parallel  to  the 
motion.                                 . 

0.4T8 

25°  33' 

0.625 

32°  1' 

0.108 

6°  10' 

6.890 

21  a  19' 

Oak  npon  oak,  fibres  of  the  moving 
body  perpendicular  to  the  motion, 

0.324 

1T.53 

0.540 

28.23 

0.143 

8^9' 

0.314 

17=26' 

Oak  upon  elm,  fibres  parallel,   . 

^0.246 

iaso 

0.376 

20.87 

0.136 

7.45 

Elm  upon  elm,    "           "                .  • 

0.140 

7.59 

Wrought  Iron  npon  oak,   .        .       .  . 

0.619 

31°47' 

0.619 

81°  47' 

"         "        "    wrought  iron,    . 

0.13S 

7.52 

0.137 

7.49 

0.177 

10.3 

u           «         «     cjjgfc              a 

0.194 

1<P59' 

0.194 

1059 

0.11S 

6.44 

«        «       «    brass,         .       /- 

0.1T2 

9.46 

0.160 

9.6 

Cast  iron  on  elm,       .... 

0.195 

11.3 

0.125 

7.S 

"       "        cast  iron,      .       . 

0.152 

8.39 

0.162 

9.13 

0.144 

8.12 

"       "        wrought  iron,      ,. 

0.143 

8.9 

«       «        brass,     .       .       .       . 

0.14T 

8.22 

0.132 

7.32 

Brass  upon  cast  iron,        .    —».      . 

0.21T 

12.15 

0.107 

6.7 

"       "      wrought  iron,        .   •    . 

0.1C1 

9.9 

"       "      brass,      ."  ,.       .       . 

0.201 

11.22 

0.154 

7.38 

0.164 

9.19 

Leather  oxhide,  well  tanned,  on  oak, 

0.296 

16.30 

•'.'*•*        "      on  cast  iron,  wetted, 

0.229 

12.54 

2.67 

1457 

"       belts  on  oaken  drums,  . 

0.27 

0.47 

"           "     "   cast  iron  pulleys,  . 

0.28 

Common  building  stones  upon  the 
same,     .                . 

0.88  to 
0.65 

20.49— 
83.2 

0.65— 
0.75 

83.2— 
36.53 

MECHANICS.  117 

From  his  experiments  M.  Morin  found,  that  the  friction  of  two  surfaces 
which  had  been  considerable  time  in  contact  was  not  only  different  in  its 
amount,  but  in  its  nature,  from  the  friction  of  surfaces  in  continuous 
motion ;  especially  in  this,  that  this  friction  of  quiescence  is  subject  to 
causes  of  variation  and  uncertainty,  from  which  the  friction  of  motion  is 
exempt;  but  that  the  slightest  jar  or  shock,  the  most  imperceptible  move- 
ment of  the  surfaces  of  contact,  was  sufficient  to  change  the  friction  from 
the  quiescent  state  into  that  which  accompanies  motion.  Hence,  as  every 
machine  or  structure  of  whatever  kind  may  be  considered  as  subject  to 
such  shock  or  imperceptible  motion,  all  questions  of  construction  depend- 
ing upon  the  state  of  friction  should  be  referred  to  that  which  accompanies 
continuous  motion.  The  friction  of  two  surfaces  outside  the  limits  of  abra- 
sion is  independent  of  the  extent  of  superficies,  and  when  in  motion,  of 
the  velocity  of  the  motion  also. 

There  are  three  states  in  respect  to  friction  into  which  the  surfaces  of 
bodies  in  contact  may  be  made  to  pass  ;  one,  a  state  in  which  no  unguent 
is  present ;  second,  a  state  in  which  the  surfaces  are  unctuous,  but  inti- 
mately in  contact ;  the  third,  a  state  in  which  the  surfaces  are  separated 
by  an  entire  stratum  of  the  interposed  unguent.  From  experiments  on 
this  last  class  Morin  deduces,  "  that  with  the  unguents  olive  oil  and  lard 
interposed  in  a  continuous  stratum  between  them,  surfaces  of  wood  on 
metal,  metal  on  wood,  wood  on  wood,  and  metal  on  metal,  when  in  motion 
have  all  of  them  very  nearly  the  same  coefficient  of  friction,  the  value  of 
that  coefficient  being  in  all  cases  included  between  0.07  and  0.08,  and  the 
limiting  angle  of  resistance  between  4°  and  4°  35'.  For  tallow  as  an' 
unguent,  the  coefficient  is  the  same  as  the  above,  except  in  case  of  metals 
upon  metals,  in  which  case  the  coefficient  was  found  to  be  0.10." 


ON  THE  EQUILIBRIUM  OF  THE  POLYGON  OF  RODS  OR  COEDS. 

If  we  take  all  the  forces  excepting  those  which  act  upon  the  extremi- 
ties of  the  polygon,  and  find  the  direction  of  their  resultant,  then  the  two 
extreme  sides  of  the  polygon  being  produced  will  meet  this  direction  in 
the  same  point.  Thus,  in  the  polygon  represented  loaded  with  the  weights 
Pl,  P2,  P3,  if  we  find  the  vertical  R  T  passing  through  the  centre  of  grav- 
ity of  these  weights,  and  produce  P  A  and  P6  B,  these  will  meet  R  T  in 
the  same  point  T  (fig.  215). 

Similarly  in  the  funicular  curve  or  catenary  (fig.  216),  if  we  draw  tan- 
gents at  the  points  of  suspension  A  and  B,  these,  being  in  the  direction  of 


118 


MECHANICS. 


the  forces  sustaining  the  curve  at  these  points,  will  meet  when  produced 
in  the  vertical  line  G  T,  passing  through  the  centre  of  gravity  G  of  the 
curve.  Let  G  T  represent  the  weight  of  the  cord  A  B  ;  draw  G  M  and 
G  Is"  parallel  to  A  T  and  B  T ;  K  T  will  represent  the  tension  at  A,  and 


Fig.  216. 


M  T  the  tension  at  B.  Such  a  curved  line  is  more  liable  to  rupture, 
therefore,  at  the  upper  point  of  suspension,  and  in  construction,  when  pos- 
sible, should  be  of  greater  dimensions. 

If  a  polygon  of  rods  be  reversed,  that  is,  placed  upright  instead  of  sus- 
pended, the  position  in  which  it  will  stand  is  that  which  it  will  assume  for 
itself  when  loaded  with  the  same  weights,  and  suspended.  Hence,  to  de- 
termine the  positions  in  which  any  number  of  beams  should  be  arranged 
in  a  polygon,  so  as  to  support  one  another,  the  timbers  of  a  gambrel  roof 
for  instance;  let  a  cord  be  taken,  and  distances  be  measured  along  it, 
equal  respectively  in  length  to  the  sides  of  the  polygon ;  let  weights  be 
attached  to  these,  equal  each  to  one  half  the  sum  of  the  weights  of  the  two 
adjacent  sides.  Then  the  two  ends  of  the  string  being  held  at  a  distance 
apart  equal  to  the  length  of  the  base  of  the  polygon,  the  form  which  the 
string  will  assume  when  hanging  freely  will  be  that  in  which  the  beams 
should  be  arranged. 

To  the  continual  equilibrium  of  an  upright  framework  it  is  essential 
that  its  joints  should  be  stiffened.  IvTow  this  cannot  be  brought  about  by 
any  peculiarity  in  the  joint  itself,  for  the  different  parts  of  such  a  joint, 
being  situated  exceedingly  near  to  the  centre  about  which  each  rod  tends 
to  move,  are,  on  the  principle  of  the  lever,  readily  crashed  by  the  action 
of  a  force,  however  slight,  acting  at  the  extremity  of  the  rod.  It  is,  there- 
fore, requisite  that  each  joint  should  be  stiffened  by  subsidiary  framing. 
And  out  of  the  necessity  for  this  strengthening  arises  the  greater  economy 
of  the  suspended  than  the  upright  polygon  or  framing.  In  the  suspended 
polygon  or  curve,  the  only  precaution  necessary  is,  that  the  parts  should 


MECHANICS. 


119 


not  tear  asunder.  In  its  uptight  position,  their  flexibility  as  well  as  the 
chance  of  their  compression,  must  be  guarded  against. 

The  methods  of  giving  rigidity  to  a  system  of  rods  are  various.  They 
all  of  them,  however,  resolve  themselves  directly  or  indirectly  into  the 
arrangement  of  the  component  rods  in  triangles.  Of  all  simple  geometri- 
cal figures,  the  triangle  is  the  only  one  which  cannot  alter  its  form  with- 
out at  the  same  time  altering  the  dimensions  of  its  sides,  and  which  cannot, 
therefore,  yield,  except  by  separating  at  its  angles,  or  tearipg  its  sides 
asunder.  Hence,  therefore,  a  triangle  whose  joints  cannot  separate,  and 
whose  sides  are  of  sufficient  strength,  is  perfectly  rigid ;  and  this  can  be 
asserted  of  no  other  plane  figure  whatever.  Thus  a  parallelogram  may 
have  sides  of  infinite  strength,  and  no  force  may  be  sufficient  to  tear  its 
joints  asunder,  and  yet  may  it  be  made  to  alter  its  form  by  the  action  of 
the  slightest  force  impressed  upon  it.  And  this  is  true  in  a  greater  or  less 
degree  of  all  other  four-sided  figures  and  polygons.  It  is  for  these  reasons 
that  in  all  framing,  care  is  taken  to  combine  all  the  parts,  as  far  as  pos- 
sible, in  triangles ;  which  being  once  done,  we  know  that  the  rigidity  of 
the  system  may  be  insured  by  giving  the  requisite  strength  to  the  timbers 
and  joints. 

The  framing  of  a  gate  presents  a  very  simple  illustration  of  this  prin- 
ciple. The  outline  of  the  form  of  the  gate  is  that  of  a  rectangular  paral- 
lelogram. If,  as  in  the  accompanying  figure  (fig.  217),  the  parts  which 
compose  it  had  been  arranged  in  directions  parallel  to  its  sides  only,  so 


Fig.  217. 


Fig.  213. 


that  the  whole  frame  should  have  been  composed  of  elementary  parallelo- 
grams, each  component  parallelogram,  and,  therefore,  the  whole  frame  of 
the  gate,  would  readily  have  altered  its  form. 

A  bar  placed  diagonally  across  the  gate  remedies  the  evil,  converting 
the  elementary  parts  of  the  gate  from  parallelograms  into  triangles,  and 
thus  giving  perfect  rigidity  to  the  frame  (fig.  218). 

Further  illustrations  of  the  principles  of  framing,  together  with  the 


120 


MECHANICS. 


equilibrium  of  solid  bodies  in  contact,  as  in  the  construction  of  retaining 
walls  and  arches,  will  be  found  under  the  head  of  Architectural  Drawing. 

THE   MECHANICAL   PROPERTIES   OF   MATERIALS. 

To  proportion  properly  the  parts  of  a  machine  or  edifice,  the  draughts- 
man should  understand  not  only  the  kind  of  strain,  the  direction,  and 
amount  to  which  its  different  parts  may  be  subjected,  but  also  the  nature 
and  properties  of  the  materials  of  which  it  may  be  composed,  and  their 
capability  of  resisting  uninjured  the  required  stress.  Many  experiments 
have  been  made  on  the  strength  of  materials,  and  in  the  tables  beneath 
will  be  seen  great  differences  in  select  specimens  of  the  same  materials,  and 
of  small  dimensions,  showing  the  necessity  of  practical  knowledge  in  selec- 
tion, and  proper  allowance  against  contingencies. 

The  forces  to  which  materials  in  constructions  are  subjected  are  com- 
pression, tension,  flexure ',  and  torsion. 

STRENGTH  OF  WOODS. — From  the  results  of  Experiments  of  Capt.  T.  J. 
Rodman,  of  the  IT.  S.  Ordnance  Department. 


HATEBIAI& 

Specific 
gravity. 

Weight  per 
cubic  foot 

Resistance  per 
To  crushing. 

square  inch. 
To  tension. 

Transverse 
resistance. 

s—  lw 

Pounds. 

Pounds. 

Pounds. 

Pounds. 

Ash,         .... 

.51  to  .73 

32  to  46 

4,500  to   8,800 

11  to  24,000 

500  to  900 

Birch,      . 

.70 

44 

8,000 

15,000 

700 

Bass                           .        . 

48  to  50 

50  to  32 

4  600  to  5  200 

12  to  15  000 

650 

Box,        .        .        .      '  .' 

.90 

56 

10,500' 

23,000 

Beech,     .... 

.67  to  .73 

42  to  46 

5,800  to  6,900 

15  to  18,000 

750 

Cedar,  red,      .        .        . 

.38 

23 

6,000 

10,000 

100 

Chestnut,         .         . 

.47  to  .54 

29  to  34 

5,100  to  5,600 

12  to  13,000 

350 

Cherry,    .         .         .         . 

.58 

36 

6,100 

12,000 

450 

Cypress,  .         .        ... 

.55 

34 

8,500 

17,000 

350 

Dogwood,        .        .        . 

.86 

54 

7,400 

23,000 

550 

Elm,         .... 

72t6.77 

45  to  48 

6,200  to  6,600 

15,000 

700 

Fir,  yellow,      .         .         . 

.55  to  .63 

34  to  39 

7,400  to  9,200 

14  to  17,000 

400  to  600 

"    red  and  white,  . 

.46 

29 

6,600  to  7,000 

13  to  15,000 

250  to  350 

Gum,  black,     . 

.61 

38 

6,700 

16,000 

500 

Hickory,  .... 

.82  to  .95 

51  to  60 

5,500  to  9,900    18  to  35,000 

900 

"      red,     .        .        . 

.72  to  .87 

45  to  54 

7,700  to  10,900 

13  to  27,000 

900  to  950 

"      white, 

.90  to.  99 

56  to  62 

8,900  to  11,200 

36  to  40,000 

950  to  1,100 

Hemlock, 

.45 

28 

6,800 

16,000 

400 

Hackmatack,   .        .        .  •'  > 

.59 

36 

450 

Lignumvitse,    .         .        . 

1.26 

79 

9,800 

16,000 

900 

Locust,    .         .         . 

.83 

52 

9,100 

27,000 

800 

Mahogany  St.  Domingo,  . 

.76 

47 

7,400 

12,000 

550 

Maple,     .... 

.6810.73 

42  to  45 

7,700  to  8,600 

22  to  23,000 

650 

Oak,  white,     .        .    ,;  . 

.63  to  .88 

39  to  55 

4,700  to  9,100  !  12  to  21,000 

500  to  950 

"     yellow,    .        .    •    P 

.71 

44 

6,300 

25,000 

600 

"     live, 

1.  to  1.1 

62  to  69 

6,500  to  7,200 

16,000 

450  to  550 

Pine,  pitch, 

1.1 

69 

8,900 

11,000 

"     white,     .        .        tf 

.36  to  .46 

23  to  29 

5,000  to  5,800 

11  to  12,000 

350 

"     yellow,    .        .        . 

.53  to  .67 

33  to  42 

7,800  to  8,400 

12  to  19,000 

500  to  650 

Poplar     . 

43  to  50 

27  to  31 

5  700  to  6  600 

8  to  15  000 

300  to  450 

Redwood,  Cal.,         .        . 

.39 

24 

6,100  ' 

11,000 

250 

Spruce,    .        .        . 

.44 

27 

5,100  to  6,800 

11  to  14,000 

350 

Teak,       .... 

.96 

60 

10,800 

31,000 

1,000 

Walnut,  black, 

.53  to  .65 

33  to  41 

5,800  to  7,500 

16  to  18,000 

450  to  650 

MECHANICS. 


121 


STRENGTH  OF  METALS. 


MATERIALS. 

Specific  . 

Weight  per 

Resistance 

per  square  inch. 

gravity. 

cubic  foot. 

To  crtishing. 

To  tension. 

Pounds. 

Pounds. 

Pounds. 

Brass,  cast, 

8.40 

523 

10,000 

18,000 

"      wire, 

49,000 

Bronze,  . 

8.70 

542 

42,000 

Copper,  cast, 

8.61 

537 

22,000 

wire, 

61,000 

Gold,  cast, 

19.26 

1,200 

20,000 

Lead,  sheet, 

11.41 

711 

3,300 

Platinum  wire 

22.07 

•  1,376 

56,000 

Silver,  cast, 

10.48 

653 

40,000 

Tin,  cast,  Banca, 

7.22 

450 

3,700 

"    wire, 

7,000 

Zinc,  cast, 

7.22 

450 

2,900 

"     sheet, 

16,000 

Iron,  cast,                                 - 

7  to  7.3 

436  to  456 

13  to  25,000 

"     wrought, 

7.6  to  7.8 

474  to  487 

20  to  110,000 

"     cable,     . 

54,  to  75,000 

"     wire, 

86  to  113,000 

Steel,  cast,  forged  bar, 

7.78 

485 

85  to  145,000 

soft,'      . 

65  to  105,000 

"      plate,     . 

41,000 

Homogeneous  metal, 

85  to  100,OOQ 

Resistance  to  Compression. — Parts  of  structures  are  usually  subjected 
to  a  compressive  force  in  the  direction  of  their  length,  and  when  this 
dimension  is  less  than  a  certain  proportion  (depending  on  the  nature  of 
the  material)  to  its  diameter  or  shortest  side,  rupture  takes  place  from  the 
absolute  crushing  of  the  parts.  In  the  actual  practice  of  construction, 
materials  cannot  with  safety  be  subjected  to  any  pressure  approaching 
their  ultimate  strength.  They  are  liable  to  various  occasional  and  acci- 
dental pressures,  and  to  others  of  a  permanent  kind,  resulting  from  settle- 
ment, and  other  causes,  of  which  no  previous  account  can  be  taken,  for 
which  allowance  must  nevertheless  be  made.  Navier,  from  existing  struc- 
tures, deduces  the  rule  that  wood  and  stone  should  not  be  subjected  to  a 
strain  over  one-tenth  of  that  which  breaks  them,  and  iron  to  not  over  one- 
fourth. 

Bricks  have  been  experimented  on  which  have  withstood  a  compressive 
force  of  13,000  Ibs.  to  the  square  inch,  whilst  others  have  failed  under  a 
pressure  of  less  than  500  Ibs. ;  and  among  the  building  stones  the  range 
would  be  even  greater.  The  granite  of  Eockfort,  Mass.,  as  tested  by  Capt. 
Eodman,  crushed  under  15,300  Ibs.  to  the  inch.  These  experiments  were 
probably  made  on  small  cubes,  and  are  of  little  use  to  the  architect ;  in 
construction  brick  is  seldom  subjected  to  a  pressure  of  100  Ibs.  per  square 
inch ;  in  fact,  few  structures  have  failed  from  an  absolute  crushing  of  either 


122  MECHANICS. 

brick  or  stone,  but  unequal  settlements  of  the  foundation  or  compression 
of  the  mortar  in  the  joints  may  change  the  strain  upon  the  material  from 
a  compressive  to  a  transverse  one  and  cause  destructive  cracks.  The 
thickness  given  for  walls  under  the  head  of  Architectural  Drawing,  will 
be  a  sufficiently  practical  guide  for  the  dimensions  of  such  works,  if 
ordinary  care  be  used  in  the  selection  of  materials  and  construction. 

Wood  and  iron,  when  used  to  resist  a  compressive  force,  are  generally 
of  such  lengths  in  comparison  with  their  sides  or  diameter,  that  rupture 
takes  place  partly  from  compression,  partly  from  flexure.  It  has  been 
found,  that  if  the  length  of  a  circular  post  exceed  eight  times  its  diameter, 
the  tendency  under  pressure  will  be  to  bend ;  and  the  longer  the  pillar, 
the  other  dimensions  remaining  the  same,  the  more  this  tendency  develops 
itself. 

For  cast-iron  posts,  the  circular  form  is  usually  adopted,  as  being  the 
strongest  form  for  the  same  amount  of  material,  the  hollow  pillar  being  in 
this  respect  preferable  to  the  solid.  Mr.  Hodgkinson,  from  his  experi- 
ments on  cast-iron  pillars,  deduces  the  following  rules : 

1st.  In  all  long  pillars  of  the  same  dimensions,  the  resistance  to  frac- 
ture by  flexure  is  three  times  greater  when  the  ends  of  the  pillar  are  flat 
and  firmly  bedded,  than  when  they  are  rounded  and  capable  of  moving. 
This  shows  the  importance  of  having  the  ends  of  pillars  turned  square,  and 
of  having  the  ends  of  braces  square  and  not  rounded,  as  has  been  proposed 
and  adopted  by  some  architects.  . 

2 d.  The  strength  of  a  pillar  with  one  end  round  and  the  other  flat,  is 
the  arithmetical  mean  between  that  of  a  pillar  of  the  same  dimensions, 
with  both  ends  rounded  and  both  ends  flat. 

Sd.  A  long  uniform  pillar,  with  its  ends  firmly  fixed,  whether  by  discs 
or  otherwise,  has  the  same  power  to  resist  breaking  as  a  pillar  of  the  same 
diameter  and  half  the  length,  with  the,  ends  rounded. 

4th.  Some  little  additional  strength  is  given  to  a  pillar  by  enlarging  its 
diameter  at  the  middle  part ;  but  this  increase  is  not  over  one-seventh  of 
the  breaking  weight. 

6th.  In  cast  iron  pillars  of  the  same  length,  the  strength  is  as  the  3.6 
power  of  the  diameter  nearly. 

6th.  In  cast  iron  pillars  of  the  same  diameter,  the  strength  is  inversely 
proportioned  to  the  1.7  power  of  the  length. 

The  breaking  weight  of  solid  cylindrical  cast  iron  pillars,  with  their 

d3'6 
ends  flat  and  incapable  of  motion,  is  in  tons  4A  x  -j-^,  I  being  the  length 

in  feet,  d  the  diameter  in  inches.     In  hollow  pillars  the  same  rule  ap- 


MECHANICS. 


123 


plies,  but  for  d™  we  use  D36  d3-6,  D  being  the  external  and  d  the  internal 
diameter.  For  pillars  with  ends  movable  and  rounded,  one-third  of  the 
above  formula  will  be  the  breaking  weight. 


TABLE  I. — Diameters  to  the  3.6  power. 


Inches. 

Inches. 

Inches. 

2. 

12.125 

5.25 

391.36 

8. 

1782.9 

2.5 

27.076 

5.5 

462.71 

8.25 

1991.7 

3. 

52.196 

5.75 

543.01 

8.5 

2217.7 

3.25 

69.628 

6. 

632.91 

8.75 

2461.7 

3.5 

90.917 

6.25 

733.11 

9. 

2724.4 

3.75 

116.55 

6.5 

844.28 

9.5 

3309.8 

4. 

147.03 

6.75 

967.15 

10. 

3981.1 

4.25 

182.89 

7. 

1102.4 

10.5 

4745.5 

4.5 

224.68 

7.25 

1250,9 

11. 

5610.7 

4.75 

272.96 

7.5 

1413.3 

11.5 

6584.3 

5. 

328.32 

-    7.75 

1590.3 

12. 

7674.5 

TABLE  II. — Lengths  to  the  1.7  power. 


Inches. 

Inches. 

Inches. 

1. 

1. 

9. 

41.900 

17. 

123.53 

2. 

3.249 

10. 

50.119 

18. 

136.13 

3. 

6.473 

11. 

58.934 

19. 

149.24 

4. 

10.556 

12. 

68.329 

20. 

162.84 

5. 

15.426 

13. 

78.289 

21. 

176.92 

6. 

21.031 

14. 

88.801 

22. 

191.48 

7. 

27.332 

15. 

99.851 

23. 

206.51 

8. 

34.297 

16. 

.  111.43 

24. 

222. 

Example. — To  find  the  breaking  weight  of  a  hollow  cast  iron  pillar 
with  square  ends,  whose  outside  diameter  is  6  inches ;  inside  or  core,  4£ ; 
and  length,  16  feet. 

Then  from  Table  I.:  — Against  6.  is  632.91;  against  4.5,  224.68. 
632.91  —  224.68  =  408.23  ;  408.23  x  44  =  17962.12;  dividing  this  by 

1  7969  1 2 
111.43,  the  number  against  16  in  Table  II,  we  have    -[-^3,  =  162  tons 

very  near  of  2,240  Ibs.  each  as  the  weight  which  would  break  this  column. 

The  above  formulas  apply  only  to  long  pillars ;  that  is,  those  whose 

length  is  at  least  25  to  30  times  their  diameter,  and  the  result  arrived  at  is 


124:  MECHAOTCS. 

their  ultimate  strength  ;  but  the  permanent  load  should  not  exceed  one- 
fourth  of  the  breaking  weight.  Short  cylindrical  pillars  may  be  loaded 
perfectly  safe  with  10  tons  to  the  square  inch  of  area  of  base.  It  is,  of 
course,  important  that  the  columns  should  be  straight  with  a  square  base, 
so  that  the  direction  of  the  strain  should  be  through  the  axis. 

Tensile  Strength.  —  Woods  are  seldom  used  to  resist  tensile  strains,  but 
when  so  used,  the  size  should  be  very  much  larger  than  that  merely  re- 
quired to  resist  the  strain,  say  at  least  ten  times.  Wrought  iron  is  the 
material  the  most  generally  employed  to  resist  a  tensile  force;  and  its 
elastic  power  in  that  capacity,  or  the  load  with  which  it  may  safely  be 
trusted,  is  ten  tons  per  square  inch  for  best  iron,  and  for  ordinary  iron, 
10,000  Ibs. 

Transverse  Strength  of  Materials.  —  The  strength  of  a  square  or  rectan- 
gular beam  to  resist  lateral  pressure,  acting  in  a  direction  perpendicular  to 
its  length,  is  as  the  breadth  and  the  square  of  the  depth  ;  and  inversely  as 
the  length,  or  the  distance  from  or  between  points  of  support.  Thus  a 
beam  twice  the  breadth  of  another,  other  proportions  being  alike,  has  twice 
the  strength  ;  or  twice  the  depth,  four  times  the  strength  ;  but  twice  the 
length,  only  half  the  strength. 

The  general  formula  is  W  —  —  j  —  ,  in  which  W  is  the  breaking  weight  ; 

{/ 

S,  a  number  determined  by  experiment  on  different  materials  (see  table,  p. 
120)  ;  b,  the  breadth,  and  d,  the  depth  in  inches  ;  and  I,  the  length  in  feet. 

To  find  the  breaking  weight  of  a  beam  supported  at  the  ends  and 
loaded  in  the  centre  :  Rule  —  Multiply  the  constant  S,  for  the  material  from 
the  above  table,  by  the  breadth  and  square  of  the  depth  in  inches,  and 
divide  the  product  by  the  length  in  feet. 

Example.  —  What  is  the  ultimate  strength  of  a  beam  of  white  pine,  20 
feet  long,  8  inches  wide,  and  14  inches  deep  ? 

S.  400  x  8  x  14'  =  627.200.     62^°°  =  31.360  Ibs. 

zo 

From  the  above  formula  we  can  determine  either  the  breadth,  depth,  or 
length,  the  other  quantities  being  known:  5  =  ^-Ta,  and  d—  /«-. 


To  determine  the  depth;  the  weight,  breadth,  and  length  being 
known  :  Multiply  the  length  in  feet  by  the  weight  in  pounds  ;  also  the 
tabular  number  S  by  the  breadth  in  inches.  Divide  the  first  product 
by  the  last,  and  the  square  root  of  the  quotient  will  be  the  required 
depth. 

To  find  the  depth  of  an  oak  timber  15  feet  long  and  6  inchel  wide,  to 


MECHANICS.  125 

support  a  weight  of  10,000  Ibs.  at  its  centre.  Multiply  the  given  weight 
by  10,  and  establish  the  depth  on  this  basis;  thus,  to  support  10,000 
securely  and  permanently  ,  find  a  beam  whose  ultimate  strength  is  equal  to 
100,000  Ibs. 

15  x  100,000  =  1,500,000.         S  700  x  6  =  4,200.     ^gff  °  =  357.1 

1/357.1  =  say  19  inches,  the  depth  required. 

When  the  load  is  not  on  the  middle  of  the  beam  :  Divide  four  times 
the  product  of  the  distance  of  the  weight  in  feet  from  each  bearing,  by 
the  whole  distance  between  the  points  of  support,  and  the  quotient  is  the 
equivalent  length  of  the  beam  loaded  in  the  middle. 

Suppose  a  beam  30  feet  in  length,  with  a  load  placed  9  feet  from  one 
end  ;  required  the  equivalent  length. 

30-9  =  21.     21  XJtX*  =  25.2  feet. 

o() 

When  the  load  is  distributed  over  the  whole  length  of  the  beam,  it  will 
bear  double  the  load  which  it  would  support  in  the  middle  ;  therefore, 
in  calculations  for  the  strength  of  a  beam  with  distributed  load,  use  double 
the  tabular  number  S  ;  if  the  ends  of  this  beam  are  firmly  fixed,  use  three 
times  S.  If  loaded  at  the  middle  with  ends  firmly  fixed,  use  1^  times  S. 

"When  a  beam  is  fixed  at  one  end,  and  the  weight  is  placed  on  the 
other  (fig.  218),  use  only  one-fourth  of  the  tabular  number  ;  if  the  load  is 
distributed  on  a  like  beam,  use  one-half  of  S. 

Ex.  —  To  find  the  depth  of  a  white  pine  beam  10  inches  wide,  project- 
ing 5  feet  from  a  wall,  and  capable  of  supporting  with  safety  2,000  Ibs. 
Call  the  "Breaking  weight  6  X  2.000  =  12.000  Ibs. 


4-00 
5  X  12.000  =  60.000.     =-x  1<T-  1.000.  «  60.     4/60  =  7.74  in. 

i  l.UUU 

It  is  only  necessary  that  the  dimensions  thus  obtained  be  preserved  at 
the  points  of  greatest  strain,  that  is,  at  A  B  (fig.  218).  i 

When  the  beam  has  two  points  of  support,  and  the  load      I  p 

is  intermediate,  the  point  of  suspension  of  the  weight  is 
tike  point  of  greatest  strain,  and  the  beam  may  be  reduced    (^\ 
towards  the  points  of  support  without  breaking  it.  Fig.  213. 

The  forms  of  beams  which  afford  equal  strength  throughout  are  para- 
bolic (figs.  219,  220,  221),  of  which  the  axis  A  B  and  the  vertex  A  are 
given,  and  the  points  M  determined  by  calculations.  Figs.  220,  221  are 
oftener  used  when  the  force  is  applied  on  alternate  sides  of  A  B. 

If  a  beam  be  subjected  to  a  transverse  strain,  one  side  is  compressed, 
while  the  other  side  is  extended  ;  and  therefore,  where  extension  terminates 


126 


MECHANICS. 


and  compression  begins,  there  is  a  lamina  or  surface  which  is  neither  ex- 
tended nor  compressed,  called  the  neutral  surface  or  neutral  axis.  As  the 
strains  are  proportional  to  the  distance  from  this  axis,  the  material  of 
which  the  beam  is  composed  should  be  concentrated  as  much  as  possible 


Fig.  219. 


at  the  outer  surface.  Acting  on  these  principles,  Mr.  Hodgkinson  has 
determined  the  most  economical  form  for  cast-iron  beams  or  girders,  of 
which  the  section  is  given  (fig.  222)  ;  it  has  been  found,  that  the  strength 
of  cast-iron  to  resist  compression  is  about  six  times  that  to  resist  exten- 
sion ;  the  top  web  is  therefore  made  only  one-sixth  the  area  of  the  lower 
one.  The  depth  of  the  beam  is  generally  about  TV  of  its  length,  the  deeper 
of  course  the  stronger ;  the  thickness  of  the  stem  or  the  upright  part 
should  be  from  1  an  inch  to  1|  inches,  according  to  the  size  of  the  beam. 
The  rule  for  finding  the  ultimate  strength  of  beams  of  the  above  section 

4 


Fig.  222. 

is : — Multiply  the  sectional  area  of  the  bottom  flange  in  square  inches  by 
the  depth  of  the  beam  in  inches,  and  divide  the  product  by  the  distance 
between  the  supports  in  feet,  and  2.16  times  the  quotient  will  be  the  break- 
ing weight  in  tons  (2240  Ibs.)  As  has  already  been  shown  above,  the  sec- 
tion thus  determined  need  only  be  that  of  the  greatest  strain,  and  can  be 
reduced  towards  the  points  of  support,  either  by  reducing  the  width  of 
the  flanges  to  a  parabolic  form  (fig.  222),  or  by  reducing  the  thickness 
of  the  bottom  flange ;  the  reduction  of  the  girder  in  depth  is  not  in  general 
as  economical  or  convenient. 

For  railway  structures  subject  to  an  impulsive  force,  the  upper  flange 


MECHANICS. 


127 


should  be  |  of  the  lower  one.  For  wrought  iron  beams,  as  this  material 
affords  less  resistance  to  compression  than  tension,  the  top  flange  is  generally 
made  larger  than  the  lower  one  in  the.proportion  of  5  to  3.  The  following 
may  be  taken  for  the  formula  to  determine  the  strength  of  solfd  wrought  iron 

beams  :    W  =  — j — ,  in  which  Tfis  the  breaking  weight,  A  the  area  of 

v 

the  section,  d  the  depth,  I  the  distance  between  the  supports.  C  is  a  con- 
stant determined  by  experiment  for  each  form  of  beam  ;  for  the  beam 
shown  in  section  (fig.  223)  C  is  found  to  be  about  40.000  Ibs. 

a  =  1.     in.  x  2.75    =  2.75  sq.  in. 

5  =  8.      "    x  0.380  =  3.04    " 

c  =  0.42  "    x  4.3      =  1.806  " 


d  =  9.42 


A  =  7.596 


Fig.  223. 


For  wrought  iron  girders  of  large  span,  the  box  form  is  generally 
adopted. 


Experiments  on  the  transverse  strength  of  rectangular  tubes  of  wrought  iron, 
supported  at  each  end,  and  the  weight  laid  on  the  middle. 


Distance  be- 
tween the 

supports. 

Breaking 
"Weight  of  tubes        weights,  exclu- 
between  the  sup-           sive  of  the 
ports.                 weights  of  the 
tubes. 

External 
depth  of  the 
tubes. 

| 

External 
breadth  of 
tubes. 

Thickness  of  the 
plates  of  the 
tubes. 

Feet. 

Tons. 

Inches. 

Inches. 

Inches. 

30.0 

42.62  cwt. 

57.5 

24 

16 

.525 

7.5 

72.36  Ibs. 

4,454 

6 

4 

.1325 

30.0 

23.09  cwt. 

22.84 

24 

16 

.272 

7.5 

35.53  Ibs. 

1.409 

6 

4 

.065 

3.75 

9.65  Ibs. 

1.1 

3 

2 

.061 

3.75 

4.34  Ibs. 

.3 

3 

2 

.03 

45.0 

130.36  cwt. 

114.76 

86 

24 

.75 

3.75 

9.65  Ibs. 

1.1 

3 

2 

.061 

30.0 

39  cwt 

54.3 

24 

16 

.50 

•A.% 


Fig.  224. 


In  several  of  these  experiments,  .the  tubes  gave  way  by 
the  metal  at  the  top  becoming  wrinkled. 

In  similar  tubes,  the  strength,  and  consequently  the 
breaking  weight,  is  proportioned  to  (1.9)  power  of  the  lineal 
dimensions. 

Approximate  formula  for  rivetted  tubes  : 

'""  \  B  D3  — •  1)  d3  1  the  breaking  weight  in  tons. 


128  MECHAIHCS. 

In  which, 

A  C  =  D,  and  a  c  =  d  in  inches ;  1)  =  length  in  feet. 
A  B  =  B,  and  a  I  =  I   «      " 

If  the  thickness  of  the  metal  be  equal  to  t  inches  completely  round 
the  section,  then  I  =  B  —  2  t,  and  d  =  D  —  2  t. 

What  is  the  breaking  weight  of  a  rectangular  tube  40  feet  long,  depth 
2  feet  6  inches,  thickness  of  plate  \  inch,  and  breadth  18  inches  ? 

(  18  x  303  —  17.5  x  29.53 


-,  486000  —  440167     =  22.96  tons. 

loUU  •( 

It  is  found  of  iron  beams  and  tubes,  that  they  may  be  safely  reduced 
in  strength  from  the  middle  towards  the  extremities  in  the  ratio  indicated 
by  theory. 

It  must  be  observed,  that  in  the  formula  given  for  determining  the 
strength  of  material,  the  force  exerted  is  supposed  to  be  dead  weight  or 
pressure,  and  that  no  consideration  is  paid  to  impulsive  force,  except  such 
slight  shocks  as  are  incident  to  all  structures.  It  is  impossible  to  give 
rules  .to  calculate  the  strength  necessary  to  resist  active  forces,  varying  in 
intensity  and  frequency  ;  we  can  only  give  instances  of  practical  structures 
which  have  been  found  sufficient,  as  mere  data  on  which  to  form  judg- 
ment.* It  must  be  remarked,  that  where  rigidity  is  required,  stiffness  of 
beams,  unlike  their  ultimate  strength,  is  directly  as  the  breadth,  and  the 
cube  of  their  depth,  and  inversely  as  the  cube  of  their  length. 

Detrusion. — The  resistance  to  detrusion,  or  the  force  necessary  to  shear 
across  any  material,  is  called  into  play  at  the  joints,  and  in  the  bolts  of 
framings  of  timber  and  iron.  The  resistance  of  spruce  to  detrusion  in  the 
direction  of  its  fibre,  is  about  600  Ibs.  per  square  inch  ;  of  cast  iron  to  de- 
trusion, about  73,000  Ibs.  per  square  inch  ;  of  wrought  iron,  45  to  50,000 
Ibs. 

Torsion. — When  two  forces  act  in  opposite  directions  upon  a  body,  tend- 
ing to  turn  its  extremities  in  different  directions,  or  twist  them,  it  is  said  to 
be  subjected  to  torsion.  Thus  the  main  shaft  of  a  steam-engine,  at  one  end 
of  which  the  power  acts  through  a  crank  ;  which  at  the  other  is  transmitted 
through  a  gear  or  pulley ;  the  resistance  which  the  load  presents  on  the 
one  hand,  and  on  the  other,  the  power  applied  to  the  crank,  represent  two 
forces,  subjecting  the  shaft  to  the  action  of  torsion. 

When  the  torsion  exceeds  a  certain  limit,  depending  on  the  material 

*  For  the  girders  of  railway  bridges,  they  should  be  of  such  dimensions  as  to  bear  a 
strain  of  two  tons  per  foot  in  length. 


MECHANICS.  129 

and  its  form,  the  fibres  arc  torn  asunder,  and  the  axles  twisted  off.  For 
the  determination  of  the  size  of  axles  subject  to  a  twisting  force,  we  de- 
duce from  "Weisbacli  the  following  rule,  allowing  a  five-fold  security  or 
strength  above  the  absolute  breaking  twist :  Multiply  the  weight  in  pounds 
by  the  leverage  in  inches,  and  divide  the  product  by  C  determined  for 
different  forms  and  material,  and  the  cube  root  of  the  product  will  be  half 

3     l-p-^ 
the  thickness  of  the  axle,  expressed  by  formula  r  =  \f  —79-. 

L/ 

Value  of  C  for  wrought  and  cast  iron,  circular  section,  12,600  Ibs. 

"  square        "        15,000   « 

"  "     wood,  circular      "  1,260   " 

"        "  square        «  1,500   " 

In  the  square  section,  the  rule  gives  as  the  cube  root  of  the  product  one- 
half  the  side  of  the  square. 

Example. — The  shaft  of  a  turbine  exerts,  through  a  toothed  wheel  of  30 
inch  pitch  or  15  inch  radius,  a  force  of  2,500  Ibs.,  what  must  be  the  diam- 
eter of  the  shaft  ? 


OKAA       -.-       O—AA  3    /37500      3     7375       1  .,  .     , 

2500  x  ID  =  3  <  oOO.        r  =  *  =  ^    —  =  1.44  inches. 


1.44  x  2  =  2.88,  say  3  inches,  diameter  required. 

The  length  of  the  axle  subjected  to  torsion  does  not  affect  the  actual 
amount  of  pressure  required  to  produce  rupture,  but  only  the  angle  of  tor- 
sion which  precedes  rupture,  and  therefore  the  space  through  which  the 
pressure  must  be  made  to  act.  The  ultimate  strength  of  a  long  shaft  to  re- 
sist torsion  may  be  sufficient,  but  its  elasticity  will  be  found  to  be  too  much. 

To  determine  the  degrees  of  the  angle  of  torsion  of  a  given  shaft,  mul- 
tiply the  load  in  pounds  by  the  leverage,  and  also  by  the  length  of  the 
shaft,  between  the  points  of  the  applied  forces  ;  and  divide  the  product  by 
the  fourth  power  of  the  half  diameter  or  half  square,  as  the  case  may  be, 
multiplied  by  a  constant  determined  by  experiment,  and  the  quotient  is 

P  a  I 
the  number  of  degrees  in  the  angle,  that  is,  ang.  =    ~    4  . 

Value  of  C. 

Circular  section.  Square  section. 

Wood,      .         .      .  .         .         3500  5800 

Cast  iron,         .         .         .     160000  280000 

Steel  and  wrought  iron,    .     280000  470000 

Example.  —  If  the  distance  of  the  toothed  wheel  from  the  water  wheel 

in  former  experiment  be  60  inches,  what  is  the  angle  of  torsion  ? 
9 


130  MECHANICS. 

37.500  x  60         375  x  6  375  x  6 


160.000  r*         160  x  1.444       160  x  4.28        4.28 

An  angle  too  considerable  for  practice ;  it  should  not  exceed  one  degree. 
To  calculate  the  size  of  the  shaft,  so  that  the  angle  is  ^  °,  the  formula  be- 

4       I  p  a  i 

comes,  r  =  \J  -^ — - ;  then  in  the  above  example, 
6  x  o- 

37.500  x  60  _  375  x  60  _  375  x  6  _ 

~80~~          ~S~      ~~  281'25 


V281.25  =  V16.77  =  4.1. 
4.1  x  2  =  8.2,  diameter  of  shaft. 

MECHANICAL    WORK    OK    EFFECT. 

To  work,  considered  in  the  abstract,  is  to  overcome,  during  any  certain 
period  of  time,  a  continuously  replaced  resistance,  or  series  of  resistances. 

Mechanical  work  is  the  effect  of  the  simple  action  of  a  force  upon  a 
resistance  which  is  directly  opposed  to  it,  and  which  it  continuously  de- 
stroys, giving  motion  in  that  direction  to  the  point  of  application  of  the 
resistance.  It  follows  from  this  definition,  that  the  mechanical  work  or 
effect  of  any  motor  is  the  product  of  two  indispensable  quantities  or  terms : 

First^ — The  effort,  or  pressure  exerted. 

Second, — The  space  passed  through  in  a  given  time,  or  the  velocity. 

The  amount  of  mechanical  work  increases  directly  as  the  increase  of 
either  of  these  terms,  and  in  the  proportion  compounded  of  the  two  when 
both  increase.  If,  f OK  example,  the  pressure  exerted  be  equal  to  4  Ibs.,  and 
the  velocity  one  foot  per  second,  the  amount  of  work  will  be  expressed  by 
4x1  =  4.  If  the  velocity  be  double,  the  work  becomes  4  x  2  =  8,  or 
double  also ;  and  if,  with  the  velocity  double,  or  2  feet  per  second,  the 
pressure  be  doubled  as  well — that  is,  raised  to  8  Ibs. — the  work  will  be 
8  x  2  =  16  Ibs.  ft. 

The  unit  of  mechanical  effect  adopted  in  England  and  this  country  is 
the  horse  power,  which  is  equal  to  33,000  Ibs.  weight  or  pressure,  raised  or 
moved  through  a  space  of  1  foot  in  a  minute  of  time.  The  corresponding 
unit  employed  in  France  is  the  kilogramme tre,  which  is  equal  to  a  kilo- 
gramme raised  one  metre  high  in  a  second.  The  horse  power  is  repre- 
sented by  75  kilogrammetres ;  that  is,  75  kilog.  raised  1  metre  high  per 
second.  When  we  speak  of  small  amounts  of  mechanical  effect,  it  is 
generally  said  that  they  are  equal  to  so  many  pounds  raised  so  many  feet 


MECHANICS.  131 

liigli  in  some  given  time,  as  a  minute  for  example.  The  time  must  always 
be  expressed  or  understood.  It  is  impossible  tp  express  or  state  intelligibly 
an  amount  of  mechanical  effect,  without  indicating  all  the  three  terms — 
pressure,  distance,  and  time. 

The  motors  generally  employed  in  manufactures  and  industrial  arts  are 
of  two  kinds — living,  as  men  and  animals  ;  and  inanimate,  as  water  and 
steam. 

AVhat  may  be  termed  the  amount  of  a  day's  work,  producible  by  men 
'and  animals,  is  the  product  of  the  force  exerted,  multiplied  into  the  distance 
or  space  passed  over,  and  the  time  during  which  the  action  is  sustained. 
There  will,  however,  in  all  cases  be  a  certain  proportion  of  effort,  in  rela- 
tion to  the  velocity  and  duration  which  will  yield  the  largest  possible  pro- 
duct or  day's  work  for  any  one  individual,  and  this  product  may  be  termed 
the  maximum  effect.  In  other  words,  a  man  will  produce  a  greater  me- 
chanical effect  by  exerting  a  certain  effort  at  a  certain  velocity,  than  he 
will  by  exerting -a  greater  effort  at  a  less  velocity,  or  a  less  effort  at  a 
greater  velocity,  and  the  proportion  of  effort  and  velocity  which  will  yield 
the  maximum  effect  is  different  in  different  individuals. 

In  the  manner  and  means  in  which  the  strength  of  men  and  animals 
is  applied,  there  are  three  circumstances  which  demand  attention  : — 

1st. — The  power,  when  the  strength  of  the  animal  is  exerted  against  a 
resistance  that  is  at  rest. 

2d. — The  power,  when  the  stationary  resistance  is  overcome,  and  the 
animal  is  in  motion.  And, 

3d. — The  power,  when  the  animal  has  attained  the  highest  amount  of 
its  speed. 

• 

In  the  first  case,  the  animal  exerts  not  only  its  muscular  force  or 
strength,  but  at  the  same  time  a  very  considerable  portion  of  its  weight  or 
gravity.  The  power,  therefore,  from  these  causes  must  be  the  greatest 
possible.  In  the  second  case,  some  portion  of  the  power  of  the  animal  is 
withdrawn  to  maintain  its  own  progressive  motion  ;  consequently  the 
amount  of  useful  labor  varies  with  the  variations  of  speed.  In  the  third 
case,  the  power  of  the  animal  is  wholly  expended  in  maintaining  its  loco- 
motion ;  it  therefore  can  carry  no  weight. 

The  following  table  exhibits  the  average  amount  of  mechanical  effect 
produced  by  men  and  animals  in  different  applications  ;  the  animal  work- 
ing with  a  mean  velocity  and  effort  during  an  average  day's  work,  thereby 
producing  the  maximum  effect. 


132 


MECHANICS. 


Nature  of  the  work. 

Effort  exerted. 

Velocily  per 
second. 

Effect  per 
second. 

Duration. 

Mechanical  effect 
per  day. 

Founds. 

Feet. 

Hoars. 

Man  -working  at  a  lever,  as  in  pump- 

tog,      

10.5 

3.5 

37.45 

8 

1.07S.560 

"    at  a  crank.    Length  of  crank  16 

to  18  inches  ;  height  of  axis  of 
shaft,  36  to  39  inches,  . 

17. 

2.4 

40.3 

8 

1.175.040 

"    tread-mill  at  level  of  axis,  . 

123. 

0.43 

61.44 

S 

1.769.472 

"       "       "    at  angle  of  24=  from 

the  vertical,        .... 

25.| 

2.25 

57.75 

8 

1.663.200 

"    at  a  vertical  capstan,  . 

25  i 

1.9 

43.45 

8 

1.395.360 

Horse  at  a  whim  gin  not  less  than  20 
feet  radius 

153. 

2.9 

443.7 

s 

12.77S.560 

Draught    bv   traces,   according    to 
Geretner  : 

.Weight. 

Man  150 

30. 

2.5 

75. 

8 

2.160.000 

Horse,  600 

120. 

4. 

430. 

8 

13.S24.000 

Mule,  500 

100. 

3.5 

350. 

3 

10.080.000 

Ox                                          600 

120. 

2.5 

300. 

8 

8.640.000 

Gerstner  also  gives  the  following  formula  to  calculate  the  effect  of 
change  of  velocity. 


K  and  c  representing  the  effort  and  velocity  per  second,  as  given  in  the 
Table,  v  the  assumed  velocity,  and  P  the  resulting  effect. 

Example.  —  Suppose  a  horse  to  travel  at  the  rate  of  6  feet  per  second. 
"What  effort  will  he  exert,  and  what  will  be  the  mechanical  effect  per 
second  ? 

From  the  table  we  have  c  =  4.  ;  K  =  120.  ;  v  is  assumed  at  6  ;  then 

/» 

P  =  (2  —  £  120  =  60.  =  effort. 
60  x  6  =  360  ft.  Ibs.  effect  per  second. 

It  is  evident  that  this  formula  will  not  apply  to  extreme  values  of  'P  or 
v  /  yet  it  may  be  considered  sufficiently  near  for  most  practical  purposes. 
not  veiy  different  from  the  mean,  and  is  illustrative  of  the  ill  effects  re- 
sulting from  increase  of  velocity. 

Water  power.  —  "Water  acts  as  a  moving  power,  or  moves  machines 
either  by  its  weight,  by  pressure,  or  by  impact  ;  and  is  applied  through 
various  forms  of  wheels.  The  mechanical  effect  inherent  in  water  is  the 
product  of  its  weight  into  the  height  from  which  it  falls  ;  but  there  are 
many  losses  incurred  in  its  application  to  machinery,  so  that  only  a  portion 
of  the  mechanical  effect  becomes  available  ;  that  is,  the  efficiency  of  any 
water  wheel  is  represented  by  a  certain  per-centage  of  the  absolute  effect 
of  the  water. 


MECHANICS.  133 

Example, — The  quantity  of  water  supplied  to  the  mills  at  Lowell  is 
3,596  cubic  feet  per  second  ;  the  net  fall  is  33  feet ;  the  absolute  dynami- 
cal effect  of  this  water  is  : 

3596  x  33  X  62.33  =  7.396.576  Ibs.  ft.  per  second. 

62.33  being  the  weight  of  a  cubic  foot  of  water,  at  60°  Fahr. ;  on  an  average 
it  may  be  assumed,  that  a  useful  effect  is  derived  equal  to  two-thirds  of 
the  total  power  of  the  water  expended ;  two-thirds  of  7.396.576,  divided 
by  550,  gives  8965.5  horse  power  as  absolutely  available.  550  Ibs.  1  foot 
high  per  second  represent  a  horse  power,  being  equal  to  33,000  Ibs.  1  foot 
high  per  minute. 

/Steam  is  the  elastic  fluid  into  which  water  is  converted  by  a  continuous 
application  of  heat.  It  is  used  to  produce  mechanical  action  almost  in- 
variably by  means  of  a  piston  movable  in  a  cylinder.  The  horse  power 
of  a  steam-engine  is  computed,  by  multiplying  the  area  of  the  piston  in 
square  inches  by  the  effective  pressure  in  Ibs.  on  each  square  inch  of  piston, 
and  the  product  by  velocity  in  feet  through  which  the  piston  moves  per 
minute,  dividing  this  last  product  by  33,000. 

The  area  of  the  piston  is  found  by  squaring  the  diameter,  and  multiply- 
ing the  square  by  0,7854. 

Example. — Let  the  diameter  of  the  piston  be  18  inches,  the  effective 
pressure  45  Ibs.  per  square  inch,  and  the  speed  300  feet  per  minute,  what 
will  be  the  horse  power  of  the  engine  ? 

18  X  18  x  0,7854  =  254.46  square  inches,  area  of  piston. 

254.46  X  45  X  300 

33,000  =104.  horse  power. 

To  determine  the  effective  pressure  on  the  piston,  recourse  must  be  had 
to  an  indicator,  and  take  the  mean  pressure,  as  shown  on  the  diagram  ;  the 
pressure  on  the  boiler  is  readily  known,  but  the  steam  in  its  passage  to  the 
cylinder  is  subject  to  various  losses,  as  of  wire-drawing,  condensation,  &c., 
so  that  frequently  the  pressure  on  the  piston  does  not  exceed  two-thirds  of 
that  on  the  boiler.  The  boilers  of  most  of  our  stationary  engines  are  sub- 
jected to  pressures  of  from  50  to  75  Ibs.  per  square  inch  ;  the  smaller  en- 
gines, say  less  than  10  horse  power,  are  generally  worked  with  full  steam ; 
effective  pressure  from  30  to  60  Ibs.  Larger  ones  are  generally  worked 
expansively,  cutting  off  at  from  one-half  to  one-sixth  stroke.  The  principle 
of  working  steam  expansively  is  as  follows :  If  a  cubic  foot  of  air  of  the 
atmospheric  density  be  compressed  into  the  compass  of  half  a  cubic  foot, 
its  elasticity  will  be  increased  from  15  Ibs.  on  the  square  inch  to  30  Ibs.  ; 
if  the  volume  be  enlarged  to  two  cubic  feet,  the  pressure  will  be  one  half, 


134 


MECHANICS. 


or  7 1  Ibs.     The  same  law  holds  in  all  other  proportions  for  gases  and  va- 
pors, provided  their  temperature  is  unchanged. 

Tims,  let  E  (fig.  225)  be  a  cy- 
linder, J  the  piston  ;  let  the  cylin- 
der be  supposed  to  be  divided  in 
the  direction  of  its  length  into  any 
number  of  equal  parts,  say  twenty, 
and  let  the  diameter  of  the  piston 
represent  the  initial  pressure  of  the 
steam,  which  we  call  1.  If  now 
the  piston  descend  through  5  of 
the  divisions,  and  the  valve  be  then 
shut,  the  pressure  at  each  subse- 
quent position  of  the  piston  may 
be  calculated  by  the  law  above 
given,  and  represented  as  shown 
in  the  figure.  If  the  squares  above 
the  point,  when  the  steam  was  cut 
off,  be  counted,  they  will  be  found 
to  amount  to  50,  those  below  to 
about  68 ;  so  that  while,  by  an 
expenditure  of  a  quarter  of  cylinder  full  of  steam,  we  get  an  amount  of 
power  represented  by  50,  we  get  68  without  any  further  expenditure,  by 
merely  permitting  expansion.  Practically,  for  large  cylinders,  it  may  be 
stated : 


Cutting  off  at 
^  stroke, 


Saves  of  fuel 
41  per  cent. 
58         " 
68         " 


Gains  in  effect 
70  per  cent. 

70  X  2  =  140  per  cent. 

70  x  3  =  210        « 


Mean  pressure  at  different  densities,  and  rate  of  expansion. 


pressure. 

EXPANSION  BY   EIGHTHS. 

i 

J 

1 

1 

i 

10 

9.896 

9.C37 

9.1  87 

8.465 

7.417 

5.965 

3.S4S 

15 

14844 

14456 

13.781 

12.697 

11.126 

8.947 

5.773 

'20 

19.792 

19.275 

18.375 

16.930 

14.835 

11.930 

7.697 

25 

24.740 

24.093 

22.963 

21.162 

18,548 

14.912 

9.621 

80 

29.6SS 

28.912 

27.562 

25.395 

22.252 

17.895 

11.546 

85 

84.636 

33.731 

33.156 

29.627 

25.961 

20.877 

13.470 

40 

39.585 

8S.550 

86.750 

83.860 

29.670 

23.860 

15.395 

45 

44.533 

43.368 

41.343 

88.099 

33.378 

26.842 

17.319 

50 

49.4S1 

4S.1S7 

45.937 

42.325 

37.067 

29.825 

19.243 

MECHANICS. 


135 


Water  converted  into  steam  under  the  pressure  of  the  atmosphere,  i.  <?., 
15  pounds  per  square  inch,  expands  to  1700  times  its  volume  ;  under  double 
the  pressure,  or  30  pounds,  the  volume  would  be  one-half;  and  this  pro- 
portion would  be  strictly  accurate  but  for  the  fact  that  the  temperatures  at 
which  water  boils  in  these  cases  are  different. 

In  the  following  table  are  given  the  total  pressure  of  steam  in  pounds 
per  square  inch,  the  corresponding  temperature,  and  the  number  of  cubic 
inches  of  steam  which  would  be  produced  by  one  cubic  inch  of  water. 


Total  pressure 
in  pounds 
per  square 
inch. 

Corresponding 
temperature. 

Cubic  inches  of     ; 
steam  produced  by 
•  a  cubic  inch  of 

Total  pressure 
in  pounds 
per  square 
inch. 

Correspond  ing 
temperature. 

Cubic  (inches  of 
steam  produced  by 
a  cubic  inch  of 
water. 

14 

209.1 

1778 

54 

288.1 

516 

15 

212.8 

1669 

55 

289.3 

503 

20 

22S.5 

1281 

56 

290.5 

500 

25 

241.0 

1044 

57 

291.7 

492 

30 

251.6 

883 

53 

292.9 

484 

35 

260.9 

761 

59 

294.2 

477 

40 

269.1 

679 

60 

295.6 

470 

45 

276.4 

610 

61 

296.9 

463 

46 

2TT.8 

598 

62 

298.1 

456 

47 

2T9.2 

586 

63 

299.2 

449 

4S 

280.5 

575 

64 

300.3 

443 

49 

231.9 

564 

65 

301.3 

437 

50 

2S3.2 

554 

66 

302.4 

431 

51 

284.4 

544 

67 

803.4 

425 

52 

285.7 

534 

68 

804.4 

419 

53 

286.9 

525 

69 

305.4 

414 

It  must  be  remarked,  that  in  non-condensing  engines,  the  effective  pres- 
sure is  the  excess  above  the  pressure  of  the  atmosphere. 

TaUe  showing  the  weights,  evaporative  powers  per  weight,  and  ~bulk  and 
character  of  fuels,  from  the  report  of  Prof.  "Walter  K.  Johnson,  1844. 


Designation  of  fuel. 

Specific 
gravity. 

Weight 
per  cubic 
foot. 

Water 
evaporst'd 
by  one  Ib. 
of  fuel. 

Designation  of  fuel. 

Specific 
gravity. 

Weight 

per  cubic 
foot. 

Water 
evaporat'd 
byonelb. 
of  fuel. 

1UTUMINOUS. 

Ibs. 

Ibs. 

ANTHRACITE. 

Ibs. 

Ibs. 

Cumberland,  maximum 

1.313 

82.09 

10.7 

Peach  Mountain, 

1.464 

91.5 

10.11 

"            minimum 

1.337 

83.28 

9.44 

Beaver  Meadow,  No.  5, 

1.554 

96.9 

9.8S 

Blossburgh, 

1.324 

82.73 

972 

Lackawana, 

1.421 

88.  S 

9.79 

Newcastle,  . 

1.257 

78.54 

S.66 

Beaver  Meadow,  No.  3,  ' 

1.610 

100.6 

9.21 

IMctou, 

1.318 

82.S3 

8.41 

Lehigh, 

1.500 

99.3 

S.93 

Pittsburgh,  . 

1252 

78.37 

8.20 

Sydney, 
Liverpool,    . 
Clover  Hill, 

1.333 
1.262 
1.2S5 

83.66 

78.89 
80.36 

7.99 
7.84 

7.  07 

Natural  Virginia, 
Cumberland, 

1.323 

82.70 

8.47 
8.99 

Cannelton,  la. 

1.273 

79/4 

7.34 

•WOOD. 

Scotch, 

1.519 

94.95 

G.95 

Dry  Pine  "Wood, 

21.01 

469 

136  MECHANICS. 

The  above  table  exhibits  the  ultimate  effects.  As  a  safe  estimate  im- 
practical values,  a  deduction  (for  the  coals)  of  TW  should  be  made. 

From  these  two  tables  it  is  easy  to  calculate  the  amount  of  fuel  which 
must  be  expended  to  produce  a  given  power. 

Example. — To  find  the  consumption  of  water  and  fuel  required  by  a 
high  pressure  engine,  12  inch  cylinder,  4  feet  stroke,  the  effective  pressure 
on  the  piston  being  40  Ibs.,  and  the  number  of  double  strokes  35  per  min. 

Area  of  piston  =  12  x  12  x  0.7854  =  113.09. 
Telocity  of  piston  =  35x8  =  280  ft.  per  min. 

Then,  113.09  X  280  X  12  =  379.982  cubic  inches  of  steam  used  during  1 
minute,  or  379.982  X  60  =  22.798.920  cub.  in.  consumption  per  hour.  Look- 
ing in  the  first  table  against  the  pressure  55,  that  is  15,  or  atmosphere 
added  to  40  given  above,  we  find  508 ;  dividing,  therefore,  22.798.920  by 
508,  we  have  44880,  the  number  of  cubic  inches  of  water  used  per  hour  ; 

44880 

~T~~28   ~  26  Cllbic  feet  nearly. 

Multiplying  this  by  the  weight  of  a  cubic  foot  of  water, 
26  x  62.33  =  1620.58  Ibs. 

Taking  the  safe  estimate  for  the  anthracites  of  the  evaporation  of  8  Ibs.  of 
water  by  1  Ib.  of  fuel,  we  have, 

—  =  202.5  Ibs.  the  consumption  of  coal  per  hour. 

o 

On  an  average  of  boilers,  1  square  foot  of  grate  surface  is  allowed  for 
the  consumption  of  14  Ibs.  of  coal  per  hour,  from  15  to  25  square  feet  of 
heating  surface,  and  one-sixth  of  a  square  foot  of  flue  at  the  base  of  the 
chimney.  Continuing  the  previous  example,  we  have 

202.5 

"--Tr—  =  14.5  square  feet  of  fire  grate. 

14.5  x  20  =  290  square  feet  of  heating  surface. 
— ^-  =  2.42  square  feet  of  flue. 

The  horse  power  of  the  above  engine  would  be 

113,09  X  280  X  40 

—  =  38.6  horse  power. 


A  portion  of  which  power  would  be  consumed  in  the  driving  of  the  engine 
itself,  leaving  about  35  horse  power  as  effective  on  the  first  shaft. 


DBA  WING   OF   MACHINERY. 


137 


DRAWING   OF  MACHINERY. 

HAVING  tlms  laid  down  the  principles  of  geometrical  projection,  and 
the  rules  by  which  to  proportion  parts,  according  to  the  stress  to  which 
they  may  be  subjected,  we  now  proceed  to  the  practical  application  ot 
the  principles  and  rules  in  the  drawing  of  machinery. 


Wood,  cast  and  wrought  iron. 


SHAFTING. 

Shafts  are  made  of  wood,  cast  and  wrought  iron.  Fig.  226  repre- 
sents the  sections  of  the  usual  forms.  Wooden  shafts  are  mostly  of  an 
octagonal  or  polyhedral  form,  and  are  seldom  used  but  as  shafts  for  water- 
wheels,  but  are  not  equal 
to  those  of  cast  iron  ex- 
cept in  cheapness,  and  are 
seldom  adopted  when  the 
latter  can  be  readily  ob- 
tained. Cast  iron  is  used 
for  the  shafts  of  water- 
wheels,  and  the  heavier 

kinds  of  mill- work,  when  rig.  220. 

the  strain  is  rather  transverse  than  torsional.  The  most  economical  form 
for  cast  iron  shafts  is  the  tubular,  but  the  more  usual  are  the  feathered 
shafts,  that  is,  with  a  circular  or  square  centre,  and  ribs  running  longi- 
tudinally. Wrought  iron  shafts  are  used  for  the  main  and  counter  shafts 
of  mills,  and  for  heavy  shafts  subject  to  torsional  or  to  unequal  stress  and 
shocks,  and  is  by  far  the  best  material  for  shafts.  The  more  usual  and  the 
best  form  is  the  circular. 

Shafts  are  termed  first,  second,  and  third  movers  ;  the  first  are  the  first 
recipients  of  power,  as  the  jack-shaft  from  a  water-wheel,  or  the  fly- wheel 
shaft  of  engines ;  the  second  are  the  next  in  succession,  distributing  the 


138 


DRAWING   OF   MACHINERY. 


power,  as  the  main  shafts  of  mills  ;  and,  third,  the  counters  or  shafts  trans- 
mitting the  power  to  the  machines.  The  strain  upon  a  shaft  may  be  trans- 
verse, torsional,  or  both.  In  all  breast,  overshot,  or  undershot  water- 
wheels,  the  jack-geer  may  be  so  placed  that  there  will  be  no  torsional 
strain  on  the  shaft  of  the  wheel ;  in  many  other  shafts,  no  strain  will  be 
transmitted  through  the  journal.  In  these  cases,  the  size  of  the  journal 
may  be  estimated  from  the  transverse  strain  or  weight  to  which  it  is  sub- 
jected. The  following  table  is  taken  from  the  Practical  Draughtsman, 
calculated  on  this  formula,  D  =  YM>  x  .1938,  D  being  the  diameter  in 
inches,  and  w  the  weight  to  be  sustained  in  Ibs. 


Table  of  the  diameters  of  the  journals  of  water-wheel  and  other  shafts  for 

heavy  work. 


• 

1 

DIAM.   OF  JOCIt 

•iAL  IN   IXCHES. 

CMt  iron. 

Wrought  iron. 

Cut  iron. 

Wrought  iron. 

1099.0 

2 

1.7 

100156 

9 

7.7 

2146.7 

2* 

2.1 

117793 

9i 

S.I 

3709.5 

3 

2.5 

1373SS 

10 

8.6 

5890.5 

^ 

3.0 

15S604 

10i 

9.0 

SS05.6 

4 

34 

1S2S64 

11 

9.4 

12019.5 

<:* 

8.8 

20S950 

1H 

9.9 

17175.5 

5 

43 

237296 

12 

10.3 

22S53.0 

t* 

4.7 

26S012 

12i 

10.7 

29676.0 

6 

5.1 

311666 

13 

11.2 

S7730.0 

Ci 

5.G 

33S026 

13* 

11.6 

43S73.0 

7 

6.0 

376993 

14 

12.0 

58915.7 

7i 

6.4 

41SS45 

1J| 

12.5 

70353.0 

's 

6.9 

4636S5 

15 

12.9 

&4373.0 

64 

7.3 

The  length  of  the  journal  should  be  from  once  to  twice  the  diameter. 
The  size  of  the  shaft  at  the  point  at  which  the  load  is  applied  may  be 
determined  from  previous  rules  ;  but  for  all  shafts  less  than  three  feet  be- 
tween bearings,  the  size  as  calculated  for  the  journal  need  only  be  enlarged 
enough  to  cut  the  key-seat. 

PL  XIV. — Figs.  1,  2,  3,  represent  different  views  of  a  wooden  water- 
wheel  shaft.  Fig.  1  shows  at  one  end  the  side  external  elevation  of  the 
shaft,  furnished  with  its  iron  ferules  or  collars  and  its  gudgeon ;  at  the 
other  end,  the  shaft  is  shown  in  sections,  giving  the  ferules  in  section, 
but  showing  the  central  spindle  with  its  feathers  in  an  external  elevation. 
Generally,  in  longitudinal  sections  of  objects  enclosing  one  or  more  pieces, 


DRAWING   OF   MACHINERY. 


139 


the  innermost  or  central  piece  is  not  sectioned  unless  it  lias  some  internal 
peculiarity,  the  object  of  a  section  being  to  show  and  explain  peculiarities, 
and  being  therefore  unnecessary  when  the  object  is  solid  ;  on  this  account, 
bolts,  nuts,  and  solid  cylindrical  shafts  are  seldom  drawn  in  section.  Fig. 
2  is  a  cross  or  transverse  section  through  the  centre  of  the  shaft,  to  show 
the  outward  octagonal  form.  Fig.  3  is  an  end  view  of  the  shaft,  showing 
the  tit-ting  of  the  spindle  B  and  its  feathers  into  the  end  of  the  shaft,  and 
the  binding  of  the  whole  by  ferules  or  hoops  a  a.  The  spindles  B,  which 
are  let  into  the  ends,  are  cast  with  four  feathers  or  wings  c.  The  tail-piece 
J  is  by  many  millwrights  omitted.  The  ends  of  the  beam  are  bored  for 
the  spindle,  and  grooved  to  receive  the  feathers  ;  the  casting  is  then  driven 
into  its  place,  hooped  with  hot  ferules,  and  after  this  hard- wood  wedges  are 
driven  in  on  each  side  of  the  feathers,  and  iron  spikes  are  sometimes  driven 
into  the  end  of  the  wood. 

Figs.  4,  5, 6,  represent  different  views  of  a  cast  iron  shaft  of  a  water-wheel. 
Fig.  4  is  an  elevation  of  the  shaft,  with  one  half  in  section  to  show  the  form 
of  the  core  ;  fig.  5,  an  end  elevation ;  fig.  6,  a  section  on  the  line  c  c  across  the 
centre.  The  body  is  cylindrical  and  hollow,  and  is  cast  with  four  feathers 
c  c,  disposed  at  right  angles  to  each  other,  and  of  an  external  parabolic  out- 
line. Kear  the  extremities  of  these  feathers  four  projections  are  cast,  for  the 
attachment  of  the  bosses  of  the  water-wheel.  These  projections  are  made 
with  facets,  so  as  to  form  the  corners  of  a  circumscribing  square,  as  shown 
in  fig.  5,  and  they  are  planed  to  receive  the  keys  by  which  they  are  fixed 
to  the  naves  which  are  grooved  to  receive  them.  The  shaft  is  cast  in  one 
entire  piece,  and  the  journals  are  turned. 

In  all  line  drawings,  the  portions  of  an  object  represented  in  section 
are  shaded  with  diagonal  lines,  as  in  figs.  1,  2,  4,  and  6. 

Fig.  227  represents  the  sec- 


tion of  a  portion  of  a  water-wheel, 
with  a  cast  iron  shaft,  in  use  in 
this  country,  in  which  stiffness  is 
given  to  the  wheel  by  wooden 
trusses,  and  a  tensional  strain  is 
given  to  the  centre  of  the  shaft. 
These  shafts  are  cast  circular  in 
two  lengths  connected  at  the  cen- 
tre, with  circular  bosses  on  which 
the  naves  of  the  wheel  are  keyed. 

When  the  load  upon  a  shaft  Fig.  227. 

is  not  central  between  the  bearings,  the  size  of  the  journals  should  be  pro- 


140 


DRAWING   OF   MACHINERY. 


portioned  to  the  weight  it  will  be  required  to  support,  which  will  be  in- 
versely as  their  distance  from  the  centre  of  pressure. 

Fig.  228  represents  the  fly-wheel  shaft  of  a  stationary  engine.     The 
parts  of  least  diameter  are  the  journals  ;  their  length  is  1^  times  the  diam- 


Fig.  22S. 


,9  "5- 


Fig.  229. 

eter  ;  the  centre  of  the  shaft  is  enlarged  to  receive  the  hub  of  the  fly-wheel, 
and  for  convenience  in  driving  the  keys.  Shafts  of  this  form  are  mostly 
of  wrought  iron,  the  reduction  being  made  by  steps,  as  a  convenience  in 
swedging.  Fig.  229  is  a  plan  of  the  crank,  from  the  wheel  side. 

The  torsional  strain  on  a  shaft  is  as  the  power  transmitted  through  it. 
It  is  evident,  power  being  weight  multiplied  by  velocity,  that  the  greater 
the  velocity  of  the  shaft,  the  less  the  strain  to  transmit  the  same  amount 
of  power ;  and  it  is  the  modem  practice  to  drive  the  shafts  at  high  veloci- 
ties, and  reduce  the  weight  of  the  geering.  In  first  movers,  the  strain  is 
often  compound  ;  and  when  the  journals  bear  but  little  transverse  strain, 
the  determination  of  their  size  must  depend  entirely  on  their  capacity  to 
resist  torsion.  The  formula  given  in  the  Practical  Draughtsman  for  de- 
termining the  proper  diameter  is  : 


C  being  for  cast  iron,  1st  movers,  419  ;  2cl,  206  ;  3d,  106. 
wr'ght"     "        "        249;    "   134;     "     67.6. 

Which  formula  is  simplified  and  tabellated,  so  that  it  is  only  necessary 
to  divide  the  number  or  revolutions  of  the  shaft  by  the  horse  power,  and 
find  the  diameter  corresponding  to  the  quotient  in  the  table. 


DRAWING   OF   MACHINERY.  141 

Table  of  diameters  for  shaft  journals,  calculated  with  reference  to  torsional 

strain* 


JOURNALS   OP  CAST-IRON   SHAFTS. 

JOURNALS  OF  WROUGHT-IRON  SHAFTS. 

Diameter  in 

inches. 

First  movers. 

Second  movers. 

Third  movers. 

First  movers. 

Second  mover. 

Third  movers. 

H 

124.133 

61.037 

81.408 

73.778 

39.704 

20.030 

2 

52.875 

25.750 

13.250 

31.125 

16.750 

8.450 

2* 

26.S1G 

13.190 

6.790 

15.372 

S.576 

4.327 

8 

15.519 

7.630 

3.922     . 

9.222 

4.963 

2.504 

*l 

9.7T3 

4.305 

2.475 

5.808 

3.123 

1.577 

4 

6.547 

3.219 

1.656 

3.S91 

2.094 

1.563 

4* 

4.593 

2.266 

1.163 

2.782 

1.475 

.742 

5  * 

3.852 

1.64S 

.343 

1.992 

1.072 

.541 

6* 

2.519 

1.239 

.637 

1.497 

.806 

.406 

6 

1.940 

.954 

.491 

1.153 

.620 

.313 

H 

1.526 

.750 

.336 

.906 

.433 

.246 

i 

1.222 

.601 

.309 

.726 

.391 

.197 

T* 

1.002 

.493 

.253 

.595 

.325 

.162 

8 

.833 

.402 

.207 

487. 

.261 

.133 

Si 

.632 

.835 

.173 

.405 

.213 

.110 

9 

.575 

.232 

.145 

.341 

.134 

.093 

9i 

.439 

.240 

.124 

.290 

.156 

.079 

10 

.419 

.206 

.106 

.249 

.134 

.063 

10i 

.362 

.173 

.092 

.215 

.116 

.058 

11 

.314 

.155 

.079 

.137 

.101 

.051 

1H 

.275 

.185 

.069 

.163 

.089 

.044 

12 

.242 

.119 

.061 

.144 

.078 

.039 

»t 

.214 

.105 

.054 

.127 

.063 

.034 

13 

.191 

.094 

.049 

.114 

.061 

.031 

181 

.170 

.034 

.043 

.101 

.054 

.027 

14 

.153 

.075 

.033 

.091 

.049 

.024 

14* 

.137 

.067 

.035 

.082 

.044 

.022 

15 

.124 

.061 

.081 

.074 

.039 

.020 

1 

2 

3 

4 

5 

6 

7 

Example,  —  "What  must  be  the  diameter  of  the  journal  of  a  wrought  iron 
first  mover,  transmitting  30  horse  power,  and  making  50  revolutions  per 
minute  ? 


1.667  in  the  table  is  intermediate  between  1.992  and  1.497,  corresponding 
to  5  and  5|,  and  should  be  about  5f  inches. 

It  is  the  common  practice  to  make  wrrought  iron  2d  and  3d  movers  of 
an  uniform  diameter,  without  reduction  at  the  journal  ;  the  shaft  is  pre- 
vented from  sliding  endways  by  collars  keyed  on.  The  usual  length  of 
main  shafts  is  from  7  to  10  feet  between  bearings  ;  and  that  they  may  run 


142  DRAWING   OF   MACHINERY. 

smooth,  and  not  spring  intermediately,  it  is  desirable  that  they  should 
never  be  less  than  2  inches  diameter,  and  that  the  pulleys  or  geers  through 
which  the  power  is  transmitted  to  the  next  mover  or  to  the  machine  should 
be  as  near  as  possible  to  the  bearing. 

Tig.  230  represents  a  line  of  shafting.  A  is  an  upright  shaft ;  a  a, 
bevel-geers ;  5  5,  bearings  for  the  shaft ;  c,  coupling  or  connection  of 
the  several  pieces  of  shafting.  These  shafts  are  intended  to  be  of  wrought 
iron.  No  reduction  is  made  for  the  journal,  no  bosses  for  pulleys  or  geers. 
As  the  power  is  distributed  from  this  line  of  shafting,  the  torsional  strain 
diminishes  with  the  distance  from  the  bevel-geers  or  first  movers,  and  the 


diameter  of  each  piece  of  shafting  may  be  reduced  consecutively,  if  neces- 
sary ;  but  uniformity  will  generally  be  found  to  be  of  more  importance 
than  a  small  saving  of  iron.  The  drawing  given  is  of  a  scale  large  enough 
to  order  shafting  by,  but  the  dimension,  should  be  written  in.  It  is  often 
usual  in  the  order  to  the  machinists  merely  to  give  the  lengths  of  the  shafts 
and  diameters  as  thus  : 

.3  .S  .2 

v      HN  a  f f-       y      ft  ft   HN      v      A  ft   H-*      v 

X          Q^      O    11.  X  >  i.ln    Qq  X  '   J.L.    <jq  X 

The  x  marks  represent  the  bearings  ;  the  joints  or  couplings  are  generally 
made  near  the  bearings,  and  it  is  also  usual  to  bring  the  pulleys  as  near 
the  bearings  as  possible.  It  frequently  happens,  therefore,  that  the  coup- 
ling and  pulley  are  needed  at  the  same  point ;  to  remedy  this,  as  the  posi- 
tion of  the  pulley  depends  on  the  machine  which  it  is  required  to  drive,  it 
frequently  cannot  be  moved  without  considerable  inconvenience  or  loss  of 
room ;  the  shaft  will  have,  therefore,  to  be  lengthened  or  shortened,  to 
change  position  of  coupling ;  or  better,  the  coupling  and  pulley  may  be 
made  together. 

BEARINGS   OR   SUPPORTS   FOR  THE   JOURNALS    OF   SHAFTS. 

for  upright  shafts. — Footstep  or  step  for  an  upright  shaft. —  Fig.  231 
represents  an  elevation  ;  fig.  232,  a  plan  of  the  step.  It  consists  of  a  foun- 
dation or  bed-plate  A,  a  box  B,  and  a  cap  or  socket  C.  The  plate  A  is 
firmly  fastened  to  the  base  on  which  it  rests  ;  in  the  case  of  heavy  shafts, 


DRAWING    OF    MACHINERY. 


143 


often  to  a  base  of  granite.  •  The  box  B  is  placed  on  A,  the  bearing  surface 
being  accurately  bevelled,  and  fitted  either  by  planing  or  chipping  and 


Fig.  232. 


filing  ;  5,  5, 5,  are  what  are  commonly  called  chipping-pieces,  which  are  the 
bearing  surfaces  of  the  bottom  of  B.     A  and  B  are  held  together  by  two 

Jjjt 


screws  ;  the  holes  for  these  are  cut  oblong  in  the  one  plate  at  right  angles 


144 


DRAWING   OF   MACHINERY. 


to  those  of  the  other  ;  this  admits  of  the  movement  of  the  box  in  two  direc- 
tions to  adjust  nicely  the  lateral  position  of  the  shaft,  after  which,  by 
means  of  the  screws  the  two  plates  are  clamped  firmly  to  each  other.  C, 
the  cup  or  bushing,  which  should  be  made  of  brass,  slips  into  a  socket  in 
B.  Frequently  circular  plates  of  steel  are  dropped  into  the  bottom  of 
this  cup  for  the  step  of  the  shaft.  The  cup  C,  in  case  of  its  sticking  to  the 
shaft,  will  revolve  with  the  shaft  in  the  box  B ;  if  plates  are  used,  these 
also  admit  of  movement  in  the  cup. 

Fig.  233  represents  the  elevation  of  a  bearing  for  an  upright  shaft,  in 
which  the  shaft  is  held  laterally  by  a  box  and  bracket  above  the  step.  The 
step  B  is  made  larger  than  the  shaft,  so  as  to  reduce  the  amount  of  wear 
incident  to  a  heavy  shaft.  The  end  of  the  shaft,  and  the  cup  containing 
oil,  are  shown  in  dotted  line.  The  bed-plate  A  rests  on  pillars,  between 
which  is  placed  a  pillow-block  or  bearing  for  horizontal  shaft. 

Figs.  234,  235,  represent  the  elevation  and  vertical  section  of  the  sus- 
pension bearing  used  by  Mr.  Boyden  for  the  support  of  the  shaft  of  his  tur- 
bine wheels.  It  having  been  found  difficult  to  supply  oil  to  the  step  of  such 
wheels,  it  was  thought  preferable  by  him  to  suspend  the  entire  weight  of 
wheel  and  shaft,  where  it  could  be  easily  attended  to.  The  shaft  (see  sec- 
tion) is  cut  into  necks,  which  rest  on  corresponding  projections  cast  in  the 


Fig.  284. 

box  5  /  the  spaces  in  the  box  are  made  somewhat  larger  than  the  necks  of 
the  shaft,  to  admit  of  Babbitting,  as  it  is  termed,  the  box ;  that  is,  the  shaft 
being  placed  in  its  position  in  the  box,  Babbitt,  or  some  other  soft  metal 
melted,  is  poured  in  round  the  shaft,  and  in  this  way  accurate  bearing  sur- 
faces are  obtained ;  projections  or  holes  are  made  in  the  box  to  hold  the 
metal  in  its  position.  The  box  is  suspended  by  lugs  5,  on  gimbals  c,  simi- 
lar to  those  used  for  mariners'  compasses,  which  give  a  flexible  bearing, 
so  that  the  necks  may  not  be  strained  by  a  slight  sway  of  the  shaft.  The 


DRAWING    OF   MACHINERY. 


screws  e  e  support  the  gimbals,  consequently  the  shaft  and  wheel ;  by  these 
screws  the  wheel  can  be  raised  or  lowered,  so  as  to  adjust  its  position  accu- 
rately ;  beneath  the  box  will  be  seen  a  movable  collar,  to  adjust  the  lateral 
position  of  shafts. 

Figs.  236,  237  are  the  plan  and  elevation  for  the  step,  or  rather  guide 
(as  it  bears  no  weight),  of  the  foot  of  the  shaft  of  these  same  turbines. 
The  plate  A  is  firmly  bolted  to  the  floor  of  the  wheel-pit ;  the  cushions,  C, 
holdino-  the  shaft,  are  either  wooden  or  cast  iron,  and  admit  of  lateral  ad- 


1 


Fig.  236. 


Fig.  237. 


justment  by  the  three  sets  of  set-screws.  "Wooden  steps  are  often  used  to 
support  the  shafts  of  the  smaller  horizontal  wheels  beneath  the  surface  of 
the  water  ;  the  fibres  of  the  wood  are  placed  vertically,  and  afford  a  very 
excellent  bearing  surface.  AVhen  cast  iron  or  steel  is  used  for  the  step,  it 
is  usual  to  encase  the  box,  and  supply  oil  by  leading  a  pipe,  sufficiently  high 
above  the  surface  of  the  water,  to  force  the  oil  down. 

For  long  upright  shafts,  it  is  very  usual  to  suspend  the  upper  portion 
by  a  suspension  box,  and  to  run  the  lower  on  a  step,  connecting  the  two 
portions  by  a  loose  sleeve  or  expansion  coupling,  to  prevent  the  unequal 
mashing  of  the  bevel  wheels,  incident  to  an  alteration  of  the  length  of  shaft 
by  variations  of  temperature.  The  suspension  is  frequently  made  by  a 
single  collar  at  the  top  of  the  shaft. 

When  a  horizontal  shaft  is  supported  from  beneath,  its  bearing  is  usu- 
ally called  a  pillow  or  plumber-block,  or  standard;  if  suspended,  the  sup- 
ports are  called  hangers. 

Figs.  238,  239  are  the  elevation  and  plan  of  a  pillow-block.  It  consists 
of  a  base  plate  A,  the  body  of  the  block  B,  and  the  box  C.  The  plate,  as 
in  the  step,  is  bolted  securely  to  its  base  ;  the  surface  on  which  the  block 
B  rests  being  horizontal.  A  and  B  are  connected  by  bolts  passing  through 

oblong  holes,  so  as  to  adjust  the  position  in  either  direction  laterally.     The 
10 


140 


DRAWING   OF   MACHINERY. 


box  or  bush  C  is  of  brass,  in  two  parts  or  halves,  extending  through  the 
block,  and  forming  a  collar  by  which  it  is  retained  in  its  place.    The  cap  of 


Fig.  239. 


the  block  is  retained  by  the  screws  o  o  o  ;  in  the  figure  there  are  two  screws 
on  one  side  and  one  on  the  other ;  often  four  are  used,  two  on  each  side, 
but  most  frequently  but  one  on  each  side. 


PROJECTIONS    OF   A   STANDAKD. 

PI.  XV. — The  standard  is  simply  a  modification  of  the  pillow-block, 
being  employed  for  the  support  of  horizontal  shafts  at  a  considerable  dis- 
tance above  the  foundation-plate.  Fig.  1  is  a  front  elevation ;  fig.  2,  a 
plan ;  and  fig.  3,  an  end  elevation  of  a  standard.  Like  the  pillow-block, 
the  plate  A  is  fastened  to  the  foundation  itself,  and  the  upper  surface  is 
placed  perfectly  level  in  both  directions.  On  these  bearing  surfaces  a  a  a 
the  body  of  the  standard  rests,  and  can  be  adjusted  in  position  horizontally, 
and  then  clamped  by  screws  to  the  foundation-plate,  or  keyed  at  the  ends. 
Fig.  4  is  a  plan  of  the  upper  part  of  the  standard  with  the  cover  off,  show- 
ing the  form  of  the  box,  with  a  babitted  -bearing  surface. 

Whilst  drawing  the  front  elevation,  mark  off  on  figs.  3  and  4  the  out- 


DRAWING-    OF   MACHINERY. 


147 


lines  of  all  such  parts  as  arc  immediately  transferable  by  the  help  of  the 
square  and  compasses,  from  one  figure  to  the  other.  The  outline  e  k  and 
fl  are  arcs,  whose  centres  lie  in  ef  produced,  and  pass  through  the  points 
e,  k  and/1,  Z.  To  find  the  projection  of  this  arc  upon  the  plan  (fig.  2),  draw 
through  any  points  m  and  n,  taken  at  pleasure  upon  the  arc  c  k  (fig.  3),  the 
horizontals  m  m,  n  n,  and  through  m  and  n  (fig.  1)  draw  m  m  and  n  n  paral- 
lel to  C  D,  then  set  off  the  distances  o  m  and^  n  (fig.  3)  to  the  corresponding 
points  on  the  lower  side  of  centre  line  M  X  (fig.  2) :  thus  the  curve  emnk 
will  be  determined.  By  a  similar  method  the  curve  c  m,  n'  will  be  ob- 
tained, as  also  the  projections  of  all  such  arcs  as  are  denoted  by  rq  (fig.  3). 

To  draw  on  fig.  3  the  s  t  h  (fig.  2),  which  is  the  line  of  penetration  of 
two  cylinders,  a  similar  construction  to  the  preceding  may  be  adopted. 
But  to  avoid  drawing  too  many  lines  on  the  figures,  this  projection  is  con- 
structed (see  fig.  5)  on  another  part  of  the  sheet,  in  which  s  t  h'  represent 
the  plan  of  the  curve  s  t  h  (fig.  2),  and  h  v*  t  the  elevation,  as  at  fig.  1. 
Divide  h  v*  t  into  any  number  of  equal  parts  ;  let  fall  perpendiculars  h  h' 
v2  vf .  .  .  from  the  points  of  division,  and  horizontal  and  parallel  lines  li  h, 
v2  v  .  .  .  ;  lay  off  on  each  side  from  the  half  chords  made  on  the  semicircle, 
and  we  have  the  curve  h  v  t  v  s,  which  may  easily  be  transferred  to  its 
position  in  fig.  3. 

It  will  be  observed,  that  one  side  of  the  elevation  (fig.  1)  is  represented 
as  broken  ;  this  is  often  done  in  drawing,  when  the  sides  are  uniform,  and 
economy  of  space  on  the  paper  is  required. 


Fig.  240.  Fig.  241. 

Suspended  bearings  or  hangers  for  horizontal  shafts  are  divided  into 
two  general   classes  —  side  hangers  (figs.  240,  241),  and  sprawl  hangers 


148  DRAWING    OF   MACHINERY. 

(pi.  XVL,  fig.  1) ;  the  figures  will  sufficiently  explain  the  distinction.  The 
side  hanger  is  the  more  convenient  when  it  is  required  to  remove  the  shaft, 
and  when  the  strain  is  in  one  direction,  against  the  upright  part ;  they  are 
generally  used  for  the  smaller  shafts,  but  sprawl  hangers  affording  a  more 
firm  support  in  both  directions,  are  used  as  supports  for  all  the  heavier 
shafts.  Hangers  are  bolted  to  the  floor  timbers,  or  to  strips  placed  to  sus- 
tain them,  the  centres  of  the  boxes  being  placed  accurately  in  line,  both 
horizontally  and  laterally. 

PI.  XYI. — Fig.  1  represents  the  elevation  of  a  sprawl  hanger ;  fig.  2, 
the  plan  looking  from  above,  with  cover  of  box  off ;  fig.  3,  a  section  on 
the  line  A  B,  fig.  1. 

Fig.  4  represents  the  elevation  of  a  bracket,  or  the  support  of  a  shaft 
bolted  to  an  upright ;  the  box  is  movable,  and  is  adjusted  laterally  by  the 
set-screws.  Fig.  5  is  a  front  elevation  of  the  back  plate  cast  on  the  post ; 
it  will  be  seen  that  the  holes  are  oblong,  to  admit  of  the  vertical  adjust- 
ment of  the  bracket.  Fig.  6  is  a  side  elevation  of  the  box  •  fig.  7,  a  sec- 
tion lengthways,  showing  aperture  for  grease,  and  the  points  which  retain 
the  babbit-metal  lining  in  its  place ;  fig.  8  is  a  plan  of  the  bottom  half  of 
the  box  ;  fig.  9,  plan  of  the  top. 

Fig.  2-42  represents  different  views  of  what  may  be  called  a  yoke- 
hanger.  Fig.  1  is  a  front  and  fig.  2  a  side  elevation ;  fig.  3  a  plan  of  the 
hanger,  looking  up ;  and  fig.  4  a  plan  of  the  yoke,  looking  down  upon  it. 
A  is  the  plate  which  is  fastened  to  the  beam,  E  is  the  yoke,  and  B  the  stem 
of  the  yoke,  cut  with  a  thread  so  as  to  admit  of  a  vertical  adjustment ;  the 
box  D  of  the  shaft  C  is  supported  by  two  pointed  set-screws  passing  through 
the  jaws  of  the  yoke ;  this  affords  a  very  flexible  bearing,  and  a  chance  for 
lateral  adjustment. 

Couplings  are  the  connections  of  shafts,  and  are  varied  in  their  con- 
struction and  proportions  often  according  to  the  mere  whim  of  the  me- 
chanic making  them. 

The  Face  Coupling  (fig.  243)  is  the  one  in  most  general  use  for  the  con- 
necting of  wrought  iron  shafts  ;  it  consists  of  two  plates  or  discs  with  long 
strong  hubs,  through  the  centre  of  which  holes  are  accurately  drilled  to 
fit  the  shaft ;  one-half  is  now  drawn  on  to  the  shaft,  and  tightly  keyed  ; 
the  plates  are  faced  square  with  the  shaft,  and  the  two  faces  are  brought 
together  by  bolts.  The  number  and  size  of  the  bolts  depend  upon  the  size 
of  the  shaft,  never  less  than  4  for  shafts  less  than  3  inches  diameter,  and 
more  as  the  diameter  increases  ;  the  size  of  the  bolts  varies  from  £  to  1  \  in. 
in  diameter.  The  figure  shows  a  usual  proportion  of  parts  for  shafts  of 
from  2  to  5  inches  diameter ;  for  larger  than  these,  the  proportion  of  the 
diameter  of  the  disc  to  that  of  the  shaft  is  too  large. 


DRAWING   OF   MACHINERY. 


149 


Fig.  244  is  a  rigid  sleeve  coupling  for  a  cast  iron  shaft ;  it  consists  of  a 
solid  1mb  or  ring  of  cast  iron  hooped  with  wrought  iron ;  the  shafts  are 
made  with  bosses,  the  coupling  is  slipped  on  to  one  of  the  shafts,  the  ends 


Fig.  242. 


of  the  two  are  then  brought  together ;  the  coupling  is  now  slipped  back 
over  the  joint,  and  firmly  keyed.     This  is  an  extremely  rigid  connection. 


Fig.  243. 


Fig.  244. 


Fig.  245  is  a  screw  coupling,  a  very  neat  and  excellent  rigid  coupling, 
for  the  connecting  of  wrought  iron,  more  especially  the  lighter  kinds.  It 
will  be  observed  that  this  coupling  admits  of  rotation  but  in  one  direction, 


150 


DRAWING    OF    MACHINERY. 


the  one  tending  to  bring  the  ends  of  the  shafts  towards  each  other,  the 
reverse  motion  tends  to  unscrew  and  throw  them  apart,  and  uncouple 
them. 

Fig.  246  is  a  clamp  coupling  for  a  square  shaft. 

In  many  cases  it  occurs  that  rigid  couplings,  such  as  we  have  given, 
are  objectionable  ;  they  necessarily  imply  that,  to  run  with  the  least  strain 


Fig.  245. 


Fig.  246. 


possible,  the  bearings  should  be  in  accurate  line ;  any  displacement  involves 
the  springing  of  the  shaft,  and  if  considerably  moved,  fracture  of  shaft  or 
coupling.  Wherever,  then,  from  any  cause  the  allignment  cannot  be  very 
nearly  accurate,  some  coupling  that,  admits  of  lateral  movement  should  be 
adopted.  The  simplest  of  these  is  the  box  or  sleeve  coupling  (tig.  247), 
sliding  over  the  end  of  two  square  shafts,  keyed  to  neither,  but  often  held 
I 


Fig.  243. 


in  place  by  a  pin  passing  through  the  coupling  into  one  of  the  shafts.  For 
round  shafts,  the  loose  sleeve  coupling  is  a  pipe  or  hub,  generally  4  to  6 
times  the  diameter  of  the  shaft  in  length,  sliding  on  keys  fixed  on  either 
shaft. 

Fig.  248  represents  a  horned  coupling.  The  two  parts  of  the  coupling 
are  counterparts  of  each  other,  each  firmly  keyed  to  its  respective  shaft, 
but  not  fastened  to  each  other ;  the  horns  of  the  one  slip  into  the  spaces 


DRAWING    OF    MACHINERY. 


151 


of  the  other ;  if  the  faces  of  the  horns  are  accurately  fitted,  it  affords  an 
excellent  coupling,  and  is  not  perfectly  rigid. 

It  often  happens  that  some  portion  of  a  shaft  or  machine  is  required  to 
be  stopped  whilst  the  rest  of  the  machinery  continues  in  motion.  It  is 
evident  that,  if  one  half  of  a  horned  coupling  be  not  keyed  to  the  shaft,  but 
permitted  to  slide  lengthways  on  the  key, — the  key  being  fixed  in  the 
shaft,  forming  in  this  case  what  is  more  usually  called  a  feather, — by  slid- 
ing back  the  half  till  the  horns  are  entirely  out  of  the  spaces  of  the  other 
half,  communication  of  motion  will  cease  from  one  shaft  to  the  other. 

Couplings  are  made  on  this  principle,  called  slide  or  clutch  couplings. 
As  usually  the  motion  is  required  but  in  one  direction,  the  more  general 
form  of  this  coupling  is  given  in  fig.  249.  A  represents  the  half  of  the 
coupling  that  is  keyed  to  the  shaft,  B  the  sliding  half,  c  the  handle  or  lever 
which  communicates  the  sliding  movement ;  the  upper  end  of  the  lever 
terminates  in  a  fork,  enclosing  the  hub  of  the  coupling,  and  fastened  by 
two  bolts  or  pins  to  a  collar  c'  round  the  neck  of  the  hub  ;  1)  is  a  box 
or  bearing  for  the  shaft  A ;  to  support  B  the  end  of  its  shaft  extends 
a  slight  distance  into  the  coupling  A.  It  will  be  observed  that  the 
horns  are  ratchet-shaped  by  this  form  motion  can  be  transmitted  but  in 
one  direction  ;  but  should  it  be  necessary  to  reverse  the  motion,  it  is  ne- 
cessary that  the  horns  of  the  coupling  be  square.  Shafts  cannot  be 


Fig.  250.  Fig.  251. 

engaged  with  this  form  of  coupling  while  the  shaft  is  in 
rapid  motion,  without  great  shock  and  injury  to  the  ma- 
chinery. To  obviate  this,  other  forms  of  coupling  are  re- 
quisite ;  one  of  these  is  represented  (fig.  250).  On  the  shaft 
B  is  fixed  a  drum  or  pulley,  which  is  embraced  by  a  friction 
band  as  tightly  as  may  be  found  necessary ;  this  band  consists  of  two  straps 
of  iron,  clamped  together  by  bolts,  leaving  ends  projecting  on  either  side  ; 
the  portion  of  the  coupling  on  the  shaft  A  is  the  common  form  of  bayonet 


152  DRAWING   OF  MACHINERY. 

clutch  ;  the  part  c  c  is  fixed  to  the  shaft,  and  affords  a  guide  to  the  prongs 
or  bayonets  1}  5,  as  they  slide  in  and  out.  Slipping  these  prongs  forward, 
they  are  thrown  into  geer  with  the  ears  of  the  friction  band  ;  the  shaft  A 
being  in  motion,  the  band  slips  round  on  its  pulley  till  the  friction  becomes 
equal  to  the  resistance,  and  the  pulley  gradually  attains  the  motion  of  the 
clutch. 

But  of  all  slide  couplings  to  engage  and  disengage  with  the  least  shock, 
and  at  any  speed,  the  friction  cone  coupling  (fig.  251)  is  by  for  the  best. 
It  consists  of  an  exterior  and  interior  cone,  a,  ~b  ;  a  is  fastened  to  the  shaft 
A,  whilst  1}  slides  in  the  usual  way  on  the  feathery  of  the  shaft  B ;  press- 
ing 5  forward,  its  exterior  surface  is  brought  in  contact  with  the  interior 
conical  surface  of  a;  this  should  be  done  gradually;  the  surfaces  of  the 
two  cones  slip  on  each  other  till  the  friction  overcomes  the  resistance,  and 
motion  is  transmitted  comparatively  gradually,  and  without  danger  to  the 
machinery.  It  must  be  observed,  that  the  longer  the  taper  of  the  cones, 
the  more  difficult  the  disengagement ;  but  the  more  blunt  the  cones,  the 
more  difficult  to  keep  the  surfaces  in  contact.  From  the  table  given,  page 
116,  it  will  be  seen  that  the  limiting  angle  of  resistance  for  surfaces  of  cast 
iron  upon  cast  iron  is  8°  39',  and  this  angle  with  the  line  of  shaft  will  give 
a  very  good  angle  for  the  surfaces  of  the  cones  of  this  material.  When 
thrown  into  geer,  the  handle  of  the  lever  or  skipper  is  slipped  into  a  notch, 
that  it  may  not  be  thrown  out  by  accident. 

Pulleys  are  used  for  the  transmission  of  motion  from  one  shaft  to 
another  by  the  means  qf  belts ;  by  them  every  change  of  velocity  may  be 
effected.  The  speed  of  the  two  shafts  will  be  to  each  other  in  the  inverse 
ratio  of  the  diameter  of  their  pulleys.  Tims,  if  the  driving  shaft  make  100 
revolutions  per  minute,  and  the  driving  pulley  be  18  inches  in  diameter, 
whilst  the  driven  pulley  is  12  inches,  then, 

12  :  18  ::  100  :  150; 

that  is,  the  driven  shaft  will  make  150  revolutions  per  minute.  Where 
there  is  a  succession  of  shafts  and  pulleys,  to  find  the  velocity  of  the  last 
driven  shaft : — Multiply  together  all  the  diameters  of  the  driving  pulleys 
by  the  speed  of  the  first  shaft,  and  divide  the  product  by  the  product  of 
the  diameters  of  all  the  driven  pulleys. 

Pulleys  are  made  of  cast  iron  and  of  every  diameter,  from  2  in.  up  to  20 
ft.  The  number  of  arms  vary  according  to  the  diameter ;  for  less  than  8  in. 
diameter  the  plate  pulley  is  preferable  (fig.  252) ;  that  is,  the  rim  is  attached 
to  the  hub  by  a  plate ;  for  pulleys  of  larger  diameters,  those  witli  arms 
are  used,  never  less  than  4  in  number.  The  arms  are  made  either  straight 


DRAWING    OF   MACHINERY. 


153 


(fig.  253),  or  curved  (fig.  254).     When  large  pulleys  are  cast  entire,  it  is 
better  that  the  arms  should  be  curved  to  admit  of  contraction  in 
for  the  smaller  it  is  unimportant. 


Fig.  252. 


Fiz.  253. 


Fig.  254. 


Fig.  255  represents  a  portion  of  the  elevation  of  a  pulley  sufficient  to 
show  the  proportion  of  the  several  parts,  and  fig.  256  a  section  of  the  same. 

Fig.  255. 


Fig.  250. 

The  parts  may  be  compared  proportionately  with  the  diameter  of  shaft ; 
thus  the  thickness  of  the  hub  is  about  \  the  diameter  of  the  shaft,  this  pro- 
portion is  also  used  for  the  hubs  of  couplings  ;  the  width  of  the  arms  from 
|  to  full  diameter  ;  the  thickness  half  the  width  ;  the  thickness  of  the  rim 
from  i  to  ~  the  diameter  ;  the  length  of  hub  the 
same  as  the  width  of  face. 

Fig.  257  represents  a  faced  coupling  pulley, 
an  expedient  sometimes  adopted  when  a  joint  oc- 
curs where  a  pulley  is  also  required,  the  two  are 
then  combined  ;  the  pulley  is  cast  in  halves — two  plate  pulleys,  with  plates 
at  the  side  instead  of  central,  faced  and  bolted  together. 


154 


DB  AWING    OF   MACHINERY. 


TTooden  pulleys  are  commonly  called  drums ;  these  are  now  but  sel- 
dom used  except  for  pulleys  of  very  wide  face.  Fig.  258  represents  one 
form  of  construction  in  elevation  and  longitudinal  section.  It  consists  of 


Fig.  258. 

two  cast  iron  pulleys  A  A,  or  spiders,  with  narrow  rims  ;  they  are  keyed 
on  to  the  shaft  at  the  required  distance  from  each  other,  and  plank  or  lag- 
ging is  bolted  on  the  rims  to  form  the  face  of  the  drum ;  the  heads  of  the 
bolts  are  sunk  beneath  the  surface  of  the  lagging,  and  the  face  is  turned. 

Fig.  259  represents  a  wooden  pulley  which  may  be  termed  a  wooden 
plate  pulley.  The  plate  consists  of  sectors  of  inch  boards  firmly  glued  and 
nailed  together,  the  joints  of  the  boards  being  always  broken.  The  face 
is  then  formed  in  a  similar  way,  by  nailing  and  gluing  arcs  of  board  one 
to  another  to  the  required  width  of  face ;  these  last  should  be  of  clear 
stuff.  The  whole  is  retained  on  the  shaft  by  an  iron  hub,  cast  with  a  plate 
on  one  side,  and  another  separate  plate  sliding  on  to  the  hub  ;  the  hub  is 
placed  in  the  centre  of  the  pulley,  the  two  plates  are  brought  in  contact 


Fig.  259. 


Fig.  260. 


with  the  sides  of  the  pulley,  and  bolted  through  ;  the  face  of  the  pulley  is 
now  turned  in  the  lathe.  A  similar  arrangement  of  hub  is  used  for  the 
hanging  of  grindstones. 

Cone  pulleys  arc  used  to  change  the  speed  of  the  driven  shaft.     Fig. 
260  represents  a  cone  pulley  with  its  hangers  ;  on  the  machine  there  is  a 


DRAWING    OF   MACHINERY. 


155 


similar  set,  but  with  ends  reversed ;  that  is,  the  large  end  of  the  hanging 
or  driving-pulley  connects  with  the  small  end  of  the  pulley  on  the  machine. 
At  this  time  the  maximum  of  velocity  is  attained  on  the  driven  shaft ;  but 
if  the  belt  is  at  the  opposite  end,  small  pulley  on  to  a  large  one,  the  speed 
is  the  minimum,  the  speed  of  the  shafts  being  in  the  inverse  ratio  of  the 
diameters  of  their  pulleys.  By  this  arrangement  speed  may  be  varied 
within  any  required  limit.  It  is  not  necessary  that  the  two  pulleys  should 
be  counterparts  of  each  other,  but  only  that  such  proportions  should  be 
preserved,  that  the  belt  may  be  tight  on  whatever  set  it  is  placed. 

The  wddth  of  the  face  of  the  pulley  depends  upon  the  width  of  the  belt 
necessary  to  transmit  the  power  ;  it  should  exceed  by  about  half  an  inch 
on  each  side  the  width  of  the  belt  for  the  ordinary  sizes.  To  determine  the 
width  of  the  belt,  determine  first  as  near  as  possible  the  power  required  to 
be  transmitted.  The  strain  on  the  belt  is  determined  by  dividing  the 
power  to  be  transmitted  by  the  velocity  ;  thus,  if  a  belt  moving  at  a  velo- 
city of  1500  feet  per  inin.  be  required  to  transmit  5  horse  power ;  that  is, 

33000  x  5  =  165000  Ibs.  ft. ;  then  =  110  Ibs.,  the  strain  on  the  belt 

JL£)Ou 

to  convey  the  power.  In  addition  to  this  strain,  it  must  be  remarked,  that 
the  belt  is  stretched  on  the  pulleys,  so  that  it  does  not  slip  while  con- 
veying the  power.  The  strain  given  above  may  be  considered  approxi- 
mately as  the  difference  of  tension  between  the  two  sides.  Morin  gives  the 
folio  wins:  Table  to  determine  the  strain  on  each  side  of  the  belt. 


Portion  of  the 

embraced  by  the 
belt. 

VALUE  OF   K. 

New  belts  on 
wooden  drums. 

Ordinnr 
On  wooden  drums. 

y  belts 
On  iron  pulleys. 

Wet  belts  on  iron 
pulleys. 

0.20 

1.S7 

l.SO 

1.42 

1.61 

0.30 

2.57 

2.43 

1.C9 

2.05 

0.40 

3.51 

3.2G 

2.02 

2.6;) 

0.50 

4.S1 

4.8S 

2.41 

3.30 

0.60 

6.59 

5.S8 

2.87 

4.19 

O.TO 

9.00 

T.90 

8.43 

5.32 

O.SO 

12.34 

10.62 

4.  09 

6.75 

0.90 

10.90 

14.27 

4.S7 

8.57 

1.00 

1 

23.14 

19.16 

5.S1 

10.S9 

Application  of  the  table. — Find  in  the  table  the  value  of  K  according 
to  the  given  circumstances ;  from  this  number  subtract  unit  or  one,  and 
divide  the  strain  on  the  belt  to  convey  the  power  by  this  remainder,  and 
the  quotient  will  be  the  minimum  tension  or  that  on  the  slack  side.  Add 


156  DRAWING   OF   MACHINERY. 

to  this  quotient  10  per  cent,  for  friction  clue  to  shafting,  or  other  causes. 
The  tension  on  the  leading  or  tight  belt  will  be  the  above  product  added 
to  the  strain,  as  given  by  the  power  required  to  be  conveyed. 

Applying  this  to  the  example  above  of  a  strain  on  the  belt  of  110  Ibs. 
with  the  ordinary  belt  embracing  1  or  0.50  of  the  circumference,  the  value 
of  K  in  the  table  is  2,41 ;  subtract  1.,  =  1.41 ;  110  divided  by  1.41  =  78 
Ibs.; 

78  +  10  per  cent,  or  78  +  7.8  =  85.8x;the  tension  on  the  slack  belt. 
85.8  +  110  =  195.8,  the  tension  on  the  tight  belt. 

Good  belting  of  an  ordinary  thickness  of  T3g  of  an  inch  should  sustain 
a  strain  of  50  Ibs.  per  inch  of  width  without  risk,  and  without  serious  wear 
for  a  considerable  time.  Therefore,  in  the  example  above,  the  belt  mov- 
ing at- a  velocity  of  1500  feet  per  minute,  required  to  transmit  a  power  of 

195  8 
five  horses,  should  be        '   ,  or  very  nearly  4  inches  in  width. 

For  the  engaging  and  disengaging  of  a  machine,  that  is,  for  putting  into 
or  out  of  motion,  the  arrangement  of  a  fast-and-loose  pulley  is  adopted  as 
simpler  and  better  than  the  clutches  before  given.  It  consists  merely  of  two 
pulleys  in  juxtaposition  on  the  same  axis,  one  fast,  the  other  loose,  so  that 
the  belt  which  transmits  the  motion  may  be  shifted  from  one  to  the  other. 
The  face  of  the  driving  pulley,  that  is,  the  one  on  the  driving  shaft,  ought 
to  be  equal  in  width  to  that  of  both  the  fast  and  loose  pulleys.  By  making 
the  face  of  the  pulleys  slightly  convex,  the  belt  is  prevented  from  slipping 
off,  as  the  tendency  of  a  belt  is  always  to  the  larger  diameter. 

When  the  belt  is  shifted,  whilst  in  motion,  to  a  new  position  on  a  drum 
or  pulley,  or  from  fast  to  loose  pulley,  or  vice  versa,  the  lateral  pressure 
must  be  applied  on  the  advancing  side  of  the  .belt,  on  the  side  on  which 
the  belt  is  approaching  the  pulley,  and  not  on  the  side  on  which  it  is  run- 
ning off.  It  is  only  necessary  that  a  belt,  to  maintain  its  position,  should 
have  its  advancing  side  in  the  plane  of  rotation  of  that  section  of  the  pulley 
on  which  it  is  required  to  remain,  without  regard  to  the  retiring  side.  On 
this  principle,  motion  may  be  conveyed  by  belts  to  shafts  oblique  to  each 
other.  Let  A  and  B  (fig.  261)  be  two  shafts  at  right  angles  to  each  other, 

A  vertical,  B  horizontal,  so  that  the 
line  run  perpendicular  to  the  direction 
of  one  axis  is  also  perpendicular  to  the 
other,  and  let  it  be  required  to  connect 
F,B  261  them  by  pulleys  and  a  belt,  that  their 

direction  of  motion  may  be  as  shown  by  the   arrows   and  their  velocities 


DRAWING   OF   MACHINERY.  157 

as  3  of  A  to  2  of  B.  On  A  describe  the  circumference  of  the  pulley  pro- 
posed on  that  shaft ;  to  this  circumference  draw  a  tangent  a  Z>  parallel  to 
m  n,  this  line  will  be  the  projection  of  the  edge  of  the  belt  as  it  leaves  A, 
and  the  centre  of  the  belt  as  it  approaches  B ;  consequently,  lay  off  the 
pulley  b  on  each  side  of  this  line,  and  of  a  diameter  proportional  to  the 
velocity  required.  To  fix  the  position  of  the  pulley  on  A,  let  fig.  262  be 
another  view  taken  at  right  angles  to  fig.  261,  and  let  the  axis  B  have  the 
direction  of  motion  indicated  by  the  arrow,  then  the  circle  of  the  pulley 
being  described,  and  a  tangent  a'  V  drawn  to  it  perpendicular  to  the 
axis  B  as  before  determined,  the  position  of  the  pulley  on  the  shaft  A  is 
likewise 


Fig.  262. 

The  positions  of  the  two  pulleys  are  thus  fixed  in  such  a  way,  that  the 
belt  is  always  delivered  by  the  pulley  it  is  receding  from,  into  the  plane  of 
rotation  of  the  pulley  towards  which  it  is  approaching.  If  the  motion  be 
reversed,  the  belt  will  run  off ;  thus  (fig.1 263),  if  the  motion  of  the  shaft  A 
is  reversed,  the  pulley  B  must  be  placed  in  the  position  shown  by  the 
dotted  lines. 

It  is  not  an  essential  condition  that  the  shafts  should  be  at  right  angles 
to  each  other  to  have  motion  transferred  by  a  belt.  They  may  be  placed 
at  any  angle  to  each  other,  provided  the  shafts  lie  in  parallel  planes,  so 
that  the  perpendicular  drawn  to  one  axis  is  perpendicular  to  the  other.  If 
otherwise,  recourse  must  be  had  to  guide-pulleys,  which  should  be  con- 
siderably convex  on  their  face. 

Geering. — The  term  geering,  in  general  sense,  is  applied  to  all  arrange- 
ments for  the  transmission  of  power  ;  it  is  also  used  in  a  particular  sense, 
as  toothed  geering. 

Toothed  geering  may  be  divided  into  two  great  classes — spur  and  level 
wheels.  In  the  former,  the  axes  of  the  driving  and  driven  wheels  are 
parallel  to  each  other  ;  in  the  latter  they  may  be  situated  at  any  angle :  if 
of  equal  size  and  at  right  angles,  they  are  called  mitre-geei's. 

Spur  wheels,  strictly  so  called,  consist  of  wheels  of  which  the  teeth  are 
disposed  at  the  outer  periphery  of  the  wheel  (PI.  XVIIL),  in  direction  of 
radii  from  their  centres. 


158  DRAWING   OF  MACHINERY. 

Internal  geering,  in  which  the  teeth  are  disposed  in  the  interior  peri- 
phery of  the  wheel,  in  direction  of  radii  from  their  centres  (plate  XXVI.) 

Hack  geer  and  pinion  are  employed  to  convert  a  rotatory  into  a  recti- 
linear motion,  or  vice  versa.  In  this  arrangement  the  pinion  is  a  spur- 
wheel,  acting  on  teeth  placed  along  a  straight  bar  (plate  XXIV.,  fig.  1.) 

Bevel-geering,  strictly  so  called,  consists  of  toothed  wheels  formed  to 
work  together  in  different  planes,  their  teeth  being  disposed  at  an  angle  to 
the  plane  of  their  faces  (plate  XXII.) 

Trundle-geer  or  wheel  is  constructed  by  inserting  the  extremities  of  a 
certain  number  of  cylindrical  pieces,  called  staves^  into  equi-distant  holes 
formed  near  the  circumferences  of  two  parallel  plates.  The  trundle  or  lan- 
tern is  in  mill-work  made  of  wood,  and  is  very  useful  when  iron  geers  can- 
not be  easily  got  or  repaired.  The  trundle  may  be  used  either  with  a  spur 
wheel  to  transmit  motion  to  parallel  shafts,  or  with  face  or  crown  wheels, 
to  transmit  motion  to  shafts  at  right  angles  to  each  other.  Face  or  crown 
wheels  are  such  as  have  their  teeth  perpendicular  to  the  plane  of  their  faces. 
The  sides  of  the  teeth  should  be  radialj  the  outer  edges  cornered,  and  in- 
serted in  a  single  plate  or  disc,  instead  of  two  as  the  trundle. 

On  the  transmission  of  motion. — The  velocity  of  rotation  of  a  driven 
wheel  depends  on  its  relative  diameter  to,  and  the  velocity  of  the  driving 
wheel  with  which,  it  is  connected.  Thus,  if  the  diameter  of  the  driven 
wheel  be  one-half  that  of  the  driver,  then  the  driven  wheel  must  make  two 
revolutions  for  one  of  the  driver.  The  driver  is  often  called  a  leader,  the 
driven  a  follower.  Hence,  to  obtain  the  diameters  of  two  wheels  having 
the  distances  apart :  Divide  the  distances  between  their  centres  into  parts, 
inversely  proportional  to  the  number  of  revolutions  which  the  wheels  are 
to  make  in  the  same  unit  of  time.  Tims,  let  A  and  B  (fig.  264)  be  'the 
given  centres,  the  ratio  of  the.ir  velocities  being  respectively  two  and  three  ; 
if  the  line  joining  the  centres  A  and  B  be  divided  into  2  +  3  =  5  equal 
parts,  that  is,  into  as  many  equal  parts  as  there  are  units  in  the  terms  of 
the  given  ratio,  the  radius  of  the  wlieel  upon  A  will  contain  three  of  these 
parts,  and  the  radius  of  the  pinion  on  B  will  contain  the  remaining  two 
parts. 

In  determining  the  size  of  a  pair  of  bevels,  we  are  not,  however,  limited 
to  any  particular  diameters  as  when  the  axes  are  parallel ;  the  wheels  may 
be  made  of  any  convenient  sizes,  and  the  teeth  consequently  of  any  breadth, 
according  to  the  stress  they  are  intended  to  bear.  The  question  is  the  mode 
of  determining  the  relative  sizes  of  the  pair  ;  and  this  resolves  itself  into  a 
division  of  the  angle  included  between  the  two  axes  inversely  as  the  ratio  of 
their  angular  velocities.  Let  B  and  C  (fig.  265)  be  the  position  of  the  two 


DRAWING    OF    MACHINEBY. 


159 


given  axes,  and  let  them  be  prolonged  till  they  meet  in  a  point  A.    Further, 
let  it  be  required  that  C  make  seven  revolutions  while.  B  makes  four.   From 


N, 


Fig.  2C4. 


Fig.  2C5. 


any  points  D  and  E  in  the  lines  A  B,  A  C,  and  perpendicular  to  them, 
draw  D  d  and  E  e  of  lengths  (from  a  scale  of  equal  parts)  inversely  as  the 
number  of  revolutions  which  the  axes  are  severally  required  to  make  in 
the  same  unit  of  time.  Thus,  the  angular  velocity  of  axis  B  being  4  (fig. 
265),  and  that  of  the  axis  C  being  7,  the  line  D  d  must  be  dra\vn  =  7,  and 
the  line  E  e  =  4.  Then  through  d  and  e  parallel  with  the  axes  A  B  and 
A  C  draw  d  c  and  e  c  till  they  meet  in  c.  A  straight  line  drawn  from  A 
through  c  will  then,  make  the  required  division  of  the  angle  BAG,  and 
define  the  line  of  contact  of  the  two  cones,  by  means  of  which  the  two  roll- 
ing frusta  may  be  projected  at  any  convenient  distance  from  A. 

Otherwise,  haA'ing  determined  the  relative  perimeters,  diameters,  or 
radii,  of  the  pair,  then  the  lines  D  d  and  E  e  are  to  each  other  directly  as 
these  quantities. 

The  point  c  may  also  be  found  more  directly  thus  :  From  A  towards  C 
in  the  axis  A  C,  set  off  from  a  scale  as  many  equal  parts  (A/*)  as  there  are 
units  in  the  number  (7)  expressing  the  velocity  of  that  axis ;  from  the  point 
f  draw  f  c  parallel  to  A  B,  and  set  off  from  the  same  scale  as  many  parts 
(f  c}  as  there  are  units  in  the  number  (4)  expressing  the  velocity  of  the 
axis  A  B  ;  then  a  line  drawn  from  A  through  c,  as  before,  wrill  divide  the 
angle  as  required. 

The  case  in  which  the  axes  are  neither  parallel  nor  intersecting  admits 
of  solution  by  means  of  a  pair  of  bevels  upon  an  intermediate  axis,  so 
situated  as  to  meet  the  others  in  any  convenient  points. 


160  DRAWDfG   OF   MACHINERY. 

When  the  contiguity  of  the  shafts  is  such  as  to  permit  of  their  being 
connected  by  a  single  pair,  skewed  bevels  are  sometimes  employed. 

When  the  axes  are  at  right  angles  to  each  other,  and  do  not  intersect, 
the  wheel  and  screw  may  be  employed  to  connect  them.  The  velocity  of 
motion  is  in  this  arrangement  immediately  deduced  from  that  of  the  screw, 
its  number  of  threads,  and  the  number  of  teeth  in  its  geering  wheel.  Tims, 
if  it  be  required  to  transmit  the  motion  of  one  shaft  to  another  contiguous, 
and  at  right  angles  to  it — the  angular  motions  being  as  20  to  1 ;  then,  if 
the  screw  be  a  single-threaded  one,  the  wheel  must  have  20  teeth ;  but  if 
double-threaded,  the  number  of  teeth  will  be  increased  to  40,  for  2  teeth 
will  be  passed  at  every  revolution.  If  the  velocities  be  as  2  to  1,  the  con- 
dition is,  that  the  screw  have  half  as  many  threads  upon  its  barrel  as  there 
are  teeth  on  the  wheel ;  and  if  1  to  1,  the  wheel  and  screw  lose  their  dis- 
tinctive characters :  both  become  many-threaded  screws  under  the  form 
of  wheels.  Wheels  of  this  sort  may  often  be  applied  with  peculiar  advan- 
tage, especially  in  light  geering  ;  and  when  so  applied,  it  is  not  essentially 
necessary  that  the  axes  be  at  right  angles  to  each  other  any  more  than  it 
is  in  bevel-geer. 

If  the  screw  have  few  threads  compared  with  the  number  of  teeth  of 
the  wheel,  it  must  always  assume  the  position  of  driver  on  account  of 
the  obliquity  of  the  thread  to  the  axis ;  and  in  this  respect  its  action  is 
analogous  to  that  of  a  travelling  rack,  moving  endwise  one  tooth,  whilst 
the  screw  makes  one  revolution  on  its  axis. 

On  the  pitch  of  wlieels. — The  primary  object  aimed  at  in  the  construc- 
tion of  toothed-geer  is  the  uniform  transmission  of  the  power,  supposing 
that  to  be  constant  and  equal.  This  implies  that  the  one  wheel  ought  to 
conduct  the  other,  as  if  they  simply  touched  in  the  plane,  passing  through 
both  their  centres.  This  plane  is  denoted  by  the  line  A  B  in  fig.  264. 

When  this  line — which  is  usually  denominated  the  line  of  centres — is 
divided  into  two  parts,  A  c  and  B  c,  proportional  to  the  number  of  teeth 
formed  upon  the  perimeters  of  the  pinion  and  wheel,  these  two  parts 
are  proportioned  or  primitive  radii  of  the  pair ;  and  a  circle  being  de- 
scribed from  each  centre  passing  through  the  common  point  c,  limits  what 
is  called  the  pitch  line  or  circle;  that  is,  a  circle  described  from  the  centre 
A,  and  another  from  the  centre  B,  through  the  same  point,  are  called,  the 
first,  \\\Q  pitch  circle  or  pitch  line  of  the  pinion,  and  the  other  of  the  wheel. 
They  are  also  sometimes  called  the  primitive  and  proportional  circles.  If 
the  pitch  circle  be  divided  into  as  many  equal  parts  as  there  are  teeth 
to  be  given  to  the  wheel,  the  length  of  one  of  these  parts  is  termed  the 
pitch  of  the  teeth.  One  of  these  arcs  comprehends  a  complete  tooth  and 


DRAWING    OF   MACHINERY. 


161 


space,  meaning  by  space  the  hollow  opening  between  two  contiguous 
teeth.  In  bevel  and  conical  wheels,  the  pitch  circle  is  the  base  of  the 
frustum. 

fiules. — I.  To  find  the  pitch  of  the  teeth  of  a  wheel,  the  diameter  and 
number  of  teeth  being  given,  divide  the  diameter  D  (in  inches)  by  the 
number  of  teeth  N,  and  multiply  the  quo- 
tient by  3.1416  :  the  product  is  the  pitch 
in  inches  or  parts  of  an  inch. 

II.  To  find  the  diameter  of  a  wheel 
the  number   of  teeth   and   pitch   being 
given,  divide  the  pitch  by  3.1416,  and 
multiply  the  quotient  by  the  number  of 
teeth. 

III.  To  find  the  number  of  teeth,  the 
diameter  and  pitch  being  given,  divide 
3.1416  by  the  pitch,  and  multiply  the  re- 
sult by  the  diameter  in  inches. 

In  ordinary  geering,  the  pitches  most 
commonly  in  use  range  from  1  inch  to  4 
inches,  increasing  up  to  two  inches  by 
eighths,  and  beyond  by  fourths  of  an 
inch.  Below  inch  the  pitches  decrease 
by  eighths  down  to  {•  inch. 

The  rules  given  above  may  be  greatly 
simplified  by  the  use  of  the  annexed 
table,  which  will  be  found  very  conven- 
ient when  the  diameter  D  is  to  be  deter- 
mined, the  pitch  P  and  number  of  teeth 
N"  being  given ;  and  conversely,  when 
the  diameter  and  pitch  are  given,  to  find 
the  number  of  teeth. 

Ex.  1.— Given  a  wheel  of  88  teeth, 


1 

p 

D=—  x  N. 

TT 

7T 

N  =  —  x  D 
P 

Pitch  in  inches 

? 
a 

f 

inch. 

the  ilium,  in  inches, 
multiply  the  number 
of  teeth  by  the  tabu- 
lar number  answer- 
ing   to    the     given 
pitch. 

the  number  of  teeth, 
multiply    the    given 
diameter    in    inches 
by  the  tabular  num- 
»T  answering  to  the 
given  pitch. 

Values  of  P. 

Values  of  — 

Values  of  — 

p 

e 

6 

1.9095 

.5236 

e 

5 

1.5915 

,6233 

_ 

H 

1.4270 

.6931 

4 

1.2732 

.7854 

8i 

1.1141 

.8976 

t 

3 

.9547 

1.0472 

4 

2} 

.8754 

1.1333 

D- 

Bf 

.7953 

1.2566 

7 

2* 

.7135 

1.3963 

a 

2 

.6366 

1.5708 

e 

is 

.5937 

1.6755 

u 

.5570 

1,7952 

r 

if 

.5141 

1.9264 

f 

H 

.4774 

2.0944 

I 

If 

.4377 

2.2S4S 

11 

.8979 

2.5132 

I* 

.3563 

2.7926 

- 

1 

.3133 

3.1416 

1 

i 

.2785 

S.5904 

1 

i 

.2337 

4.1  8SS 

i 

.1939 

5.0266 

i 

.1592 

6.2332 

f. 

.1194 

8.3776 

i 

.0796 

12.5664 

2i-inch  pitch,  to  find  the  diameter  of  the  pitch  circle.  Here  the  tabular 
number  in  the  second  column  answering  to  the  given  pitch  is  .7958,  which 
multiplied  by  88  gives  70.03  for  the  diameter  required. 

2.  Given  a  wheel  33  inches  diameter,  If -inch  pitch,  to  find  the  number 

of  teeth.     The  corresponding  factor  is  1.7952,  which  multiplied  by  33, 

gives  59.242  for  the  number  of  teeth,  that  is,  59|  teeth  nearly.     Now  59 

would  here  be  the  nearest  whole  number,  but  as  a  wheel  of  60  teeth  may 

11 


162 


DRAWING    OF    MACHESTERY. 


be  preferred  for  convenience  of  calculation  of  speeds,  we  may  adopt  that 
number,  and  find  the  diameter  corresponding.  The  factor  in  the  second 
column  answering  to  If  pitch  is  .557,  and  this  multiplied  by  60  gives  33.4 
inches  as  the  diameter  which  the  wheel  ought  to  have. 

Another  mode  of  sizing  wheels  in.  relation  to  their  pitches,  diameters, 
and  number  of  teeth,  is  adopted  in  some  machine  shops,  by  dividing  the 
diameter  of  the  pitch  circle  into  as  many  equal  parts  as  there  are  teeth  to 
be  given  to  the  wheel.  To  illustrate  this  by  an  arithmetical  example,  let 
it  be  assumed  that  a  wheel  of  20  inches  diameter  is  required  to  have  40 
teeth  ;  then  the  diametral  pitch, 


20 

•777 
40 


that  is,  the  diameter  being  divided  into  equal  parts  corresponding  in  num- 
ber to  the  number  of  teeth  in  the  circumference  of  the  wheel,  the  length 
of  each.  of  these  parts  is  |  an  inch,  consequently  m  —  2  ;  and  according  to 
the  phraseology  of  the  workshop,  the  wheel  is  said  to  be  one  of  two  pitch. 
In  this  mode  of  sizing  wheels,  a  few  determined  values  are  given  to 
m,  as  20,  16,  14,  12,  10,  9,  8,  T,  6,  5,  4,  3,  2,  1,  which  includes  a  variety 
of  pitches  from  }  inch  up  to  3  inches,  according  to  the  following  table, 
which  shows  the  value  of  the  circular  pitches  corresponding  to  the  assigned 
values  of  m. 


Values  of  m. 

1 

2 

3 

4 

5 

1 

7 

8 

9 

10 

12 

14 

16  j  20 

Corresponding  circular  pitch  in  deci-  1 
mals  of  an  inch,                                  J 

3.142 

1.571 

1.047 

.785 

.025 

524 

.449 

m 

.849 

314 

2C2 

.•224 

196  .157 

As  it  is  convenient  to  express  all  the  dimensions  in  terms  of  the  same 
unit,  and  the  pitch  being  an  appropriate  quantity,  it  is  nearly  universally 


Fig.  266. 


adopted  as  the  term  of  comparison.  The  following  are  the  proportions 
adopted  by  different  workshops,  some  preferring  one  and  some  the  other 
(fig.  266). 


DRAWING    OF    MACHINERY.  163 

A  C  =  Pitch  of  teeth,  =    1  pitch,  or  =  15  parts. 

ac  —  Depth  to  pitch  line,  P  P,  =  T3«r  "  "  =    o*  " 

A.a  +  'ac  =  Working  depth  of  tooth,  =  TV  .  "  "  =  11  " 

Cc  —  A a  =  Bottom  clearance,  =  TO  "  "  =     1  " 

C  a  =  Whole  depth  to  root,    '  =  TV  u  "  =  12  " 

C  b  =  Tliickness  of  tooth,  =  T\  "  «  =    7  " 

A  5  =  Width  of  space,  =  -A  "  "  =    8  " 

In  practice,  these  proportions  are  often  laid  off  in  lines  for  the  conveni- 
ence of  the  workmen  in  the  pattern  shop,  so  that  for  any  given  pitch  the 
other  dimensions  may  at  once  be  determined  by  means  of  the  compasses. 
In  figs.  267  and  268,  two  diagrams  of  that  sort  are  given.  Fig.  267  con- 
tains the  proportions  last  enumerated,  in  which  the  pitch  is  supposed  to  be 
divided  into  15  equal  parts  ;  and  fig.  268  is  constructed  nearly  according 
to  the  proportions  first  given,  but  embraces  the  recognised  principle,  that 
the  relative  amount  of  clearance  ought  to  vary  inversely  as  the  pitch, 
wheels  of  small  pitch  requiring  more  clearance  relatively  than  those  in 
which  the  pitch  is  greater. 

The  construction  of  these  scales  is  very  simple.  Thus,  in  fig.  267,  let 
A  B  be  divided  into  15  equal  parts,  and  draw  B  C  perpendicular  to  it ;  and 
again  divide  B  C  into  a  determinate  number  of  parts  from  B,  actual  meas- 
ures of  the  pitches  for  which  the  scale  is  intended  to  be  used  ;  that  is,  B  a 
=  i  inch ;  B  5  =  1  inch  ;  B  c  =  2  inches,  and  so  on,  and  join  a  and  A, 
5  and  A,  c  and  A,  and  so  on.  To  complete  the  scale,  draw  15  parallels  to 
B  C  from  the  points  numbered  in  the  line  A  B,  and  also  the  two  parallels 
T  and  II  equidistant  from  the  parallels  on  each  side  of  them. 

The  scale  is  thus  ready  for  use.  To  get  from  it  the  several  proportions 
for  a  given  pitch,  say  of  three  inches  =  B  J,  let  the  compasses  be  extended 
from  the  intersection  of  the  parallel  marked  T,  with  the  line  A  B,  to  the 
point  where  it  intersects  the  line  Ad;  this  will  be  the  part  of  the  tooth 
from  the  pitch  line  to  the  point,  and  equivalent  to  5|  parts  of  the  pitch, 
(viz.,  of  B  d) ;  similarly,  the  compasses  being  extended  from  the  intersec- 
tion of  the  parallel  U  with  the  line  A  B  to  its  point  of  intersection  of  the 
line  A  d,  will  give  the  part  of  the  length  of  the  tooth  from  the  pitch  line 
to  the  root,  and  equivalent  to  6£  parts  of  the  pitch.  For  the  whole 'length 
of  the  tooth  (if  wanted  in  one  measurement),  set  the  compasses  to  the  point 
where  the  parallel  marked  12  meets  the  line  A  B,  and  extend -to  its  point 
of  intersection  of  the  line  A  d  at  *,  the  length  is  12  parts  of  the  pitch  B  d; 
the  working  depth  is  in  like  manner  found  from  the  parallel  marked  11, 
the  thickness  from  that  marked  7,  and  the  width  of  space  from  that 
marked  8. 


164 


DEAWING    OF    MACHINERY. 


The  proportions  for  any  other  given  pitch  comprised  in  the  scale  are 
found  in  precisely  the  same  way  ;  and  if  the  scale -be  well  constructed,  they 
may  be  measured  off  with  the  utmost  accuracy  and  readiness.  To  save 
confusion,  it  is,  however,  better  in  practice  to  insert  in  the  diagram  only 

Proportion  scales  for  geering. 


those  parallels,  namely,  T,  U,  12,  11,  8,  7,  which  are  required ;  the  others 
are  not  requisite,  and  by  inattention  may  lead  to  error. 

The  description  of  the  scale  as  here  given  supposes  that  the  lateral 


DRAWING    OF   MACHINERY.  165 

clearance  is  constantly  l-15th  of  the  pitcli ;  but  as  it  is  commonly  desirable 
that  this  should  vary  slightly  with  the  pitch,  relatively  increasing  as  the 
pitch  decreases,  two  other  lines,  in  n  and  p  q,  have  been  introduced  into 
the  scale,  to  enable  such  modification  to  be  adopted  should  it  be  required. 
These  lines  are  drawn  at  such  angles  as  to  give  a  clearance  at  6  inches 
pitcli  of  l-18th,  which  is  increased  at  |-inch  pitch  to  l-10th.  From  these 
lines  the  thickness  and  space  are  to  be  taken,  instead  of  using  the  lines 
marked  T  and  8,  setting  the  compasses  in  the  points  of  intersection  with 
the  pitch  lines,  and  extending  perpendicularly  to  the  line  A  B  ;  in  other 
words,  the  shortest  distance  from  the  point  of  intersection  with  the  pitch 
line  to  the  line  A  B  is  the  required  measure  of  the  space  when  the 
line  p  q  is  taken,  and  of  the  thickness  of  tooth  when  the  line  m  n  is 
taken. 

Fig.  268  is  more  complete  than  the  one  described  ;  the  principle  of  its 
construction  is  in  cifect  the  same,  but  its  use  is  more  extended,  the  diam- 
eter of  the  wheel  being  found  from  it  simultaneously  with  the  length  and 
thickness  of  tooth,  width  of  space,  and  clearances.  The  scale  is  adapted  to 
wheels  of  all  the  pitches,  from  \  inch  up  to  3  inches.  The  mode  of  con- 
struction is  this :  having  drawn  the  line  A  D  of  any  convenient  length, 
raise  the  perpendicular  C  B  to  it,  also  of  any  convenient  length.  On  the 
line  A  D  lay  off  the  greatest  pitcli  of  the  scale  from  C  to  A ;  then  from 
C  towards  D  lay  off  seven  times  the  pitch  once  or  twice,  according  to  the 
sizes  of  wheels  to  which  the  scale  is  intended  to  be  applied.  In  the  scale 
given,  double  of  seAren  times  the  pitch  is  laid  off,  namely,  42  inches  ;  then 
each  of  these  great  divisions  being  subdivided  into  11  equal  parts,  one  of 
these  parts  will  be  equal  to  four  teeth  upon  the  radius  of  the  wheel,  so  that 
the  whole  line  C  D  will  be  divided  into  88  radial  pitches.  Next  on  the 
line  C  B  set  off  the  pitches  which  may  be  required  in  the  scale,  and^ 
through  these  points  draw  the  24  parallels  to  A  D,  terminating  in  the  lines 
A  B  and  D  B.  Then  each  parallel  measured  from  the  line  B  C  to  its 
point  of  termination  in  B  D  is  the  radius  of  a  wheel  of  88  teeth  of  the  par- 
ticular pitch  marked  against  it  on  the  line  A  B.  They  also  express  the 
radii  of  wheels  having  less  than  88  teeth  when  measured  only  to  the 
corresponding  point  in  the  line  joining  B,  and  the  divisional  on  C  D, 
against  which  the  number  of  teeth  is  marked.  Thus,  the  radius  of  a 
wheel  of  52  teeth  and  If-inch  pitch  is  r  s  =  15  7-16th  inches  very  nearly. 
(The  true  answer  by  the  table,  page  161,  is  30.8724  -f-  2  =  15.4362 
inches). 

The  scale  may  also  be  used  when  the  number  of  teeth  exceeds  88  ;  for 
example,  to  find  the  radius  of  a  wheel  having  100  teeth.  Thus,  having 


166  DRAWING   OF   MACHINERY. 

found  the  radius  answering  to  88  teeth,  upon  the  same  parallel  take  off  the 
measure  answering  to  the  difference  100  —  88  =  12  teeth ;  and  the  two 
measures  together  will  be  the  radius  required. 

To  adapt  the  scale  to  odd  numbers  of  teeth,  the  first  division  on  the 
right  of  C  is  divided  into  single  radial  pitches,  so  that  the  radius  of  any 
wheel  may  be  measured  off  without  having  recourse  to  calculation  of  any 
kind.  Tims,  for  example,  if  the  wheel  is  intended  to  contain  50  teeth,  the 
parallel,  answering  to  the  particular  pitch,  comprehended  between  52  B 
and  2  B,  will  give  the  radius  required,  that  is,  a  radius  answering  to 
52  —  2  =  50  teeth  ;  and  any  other  number  of  teeth,  when  not  marked 
against  the  base,  may  be  found  in  the  same  way,  observing  that  it  is  more 
convenient  to  subtract  than  to  add  in  this  use  of  the  scale. 

For  the  proportions  of  the  te^th,  set  off  C  a  =  h( '-tenths  of  the  pitch, 
then  will  A  a  =  2>-tenths  of  the  pitch,  which  corresponds  to  the  depth  from 
the  point  of  the  tooth  to  the  pitch  line.  Again,  set  off  C  5  =  1 -fifteenth* 
of  the  3-inch  pitch,  and  ^-elevenths  on  the  parallel  against  the  1-inch  pitch ; 
this  will  be  the  thickness  of  the  tooth,  allowing  from  a  fifteenth  for  clear- 
ance on  the  largest  pitch  to  a  tenth  on  those  from  f-inch  and  under ;  and 
A  5  will  be  the  width  of  space,  including  the  clearance.  Lines  being 
drawn  from  those  points  to  B  complete  the  diagram,  which  will  be  found 
to  contain  all  the  proportions  enumerated  in  the  preceding  table. 

To  use  the  scale,  lay  off  the  addendum  of  the  tooth  ;  that  is,  the  length 
beyond  the  pitch  line,  equal  to  A  a  =  T\  pitch,  and  the  same  length 
marked  off  within  the  pitch  line  will  give  the  whole  working  depth  of  the 
tooth,  namely,  6-10ths  pitch.  Then  with  the  measure  C  a  =  T\  pitch  in 
the  compasses,  mark  off  the  whole  length  of  the  tooth,  and  this  will  allow 
l-10th  at  bottom  for  clearance.  Again,  set  off  the  thickness  of  tooth  = 
C  5,  and  the  space  =  A  5,  which  will  contain  the  clearance  for  the  par- 
ticular pitch,  varying  from  l-15th  to  fully  l-10th  on  the  small  pitches.  It 
is  hardly  necessary  to  observe,  that  these  measurements  must  be  taken 
upon  the  parallel  corresponding  to  the  particular  pitch  under  considera- 
tion at  the  time. 

The  amount  of  bottom  clearance  is  here  presumed  to  be  uniformly 
l-10th  of  the  pitch  ;  but  if  it  be  thought  advisable  to  make  this  vary,  as  in 
the  case  of  the  lateral  clearance,  it  will  then  be  necessary  to  insert  a  third 
line  c  B  in  the  scale,  and  so  related  to  a  B  that  the  space  a  c  shall  be 
throughout  equal  to  the  depth  of  tooth  from  the  pitch  circle  to  the  root, 
and  giving  any  bottom  clearance  that  may  be  desired. 

In  relation  to  the  strength  of  wheels,  M.  Morin  gives  it  as  a  rule,  that 
when  the  velocity  of  the  pitch  circle  does  not  exceed  five  feet  per  second, 


DRAWING   OF   MACHINERY. 


167 


the  breadth  of  the  tooth  measured  parallel  to  the  axis  ought  to  be  equal 
to  four  times  the  thickness  ;  but  when  the  velocity  is  higher,  the  breadth 
ought  to  be  equal  to  five  thicknesses,  the  teeth  being  constantly  greased. 
If  the  teeth  be  constantly  wet,  he  recommends  the  breadth  to  be  made 
equal  to  six  thicknesses  at  all  velocities. 

With  respect  to  the  thickness  of  the  tooth,  it  is  plain  that  it  must  be 
dependent  on  the  pressure  which  the  tooth  is  required  to  sustain,  and  upon 
the  nature  of  the  material  of  which  it  is  formed.  We  subjoin  the  follow- 
ing table  for  calculating  the  strength  of  cast  iron  wheels. 

For  teeth  of  wood  add  50  per  cent,  to 
the  thickness  as  given  by  Table. 

When  cast  iron  and  wooden  teeth 
work  together,  their  action  upon  each 
other  tends,  in  consequence  of  the  elas- 
ticity of  the  wood,  to  maintain  a  more 
uniform  distribution  of  the  strain,  and 
being  at  first  commonly  more  accurately 
dressed,  to  prevent  abrasion  of  wood  by 
the  iron,  they  work  with  much  less  fric- 
tion, are  less  liable  to  shocks,  and  nearly 
exempt  from  accident  by  hard  particles 
coming  between  the  teeth. 

The  best  practice,  when  a  mortise  and 
iron  wheel  are  to  work  together,  is  to 
make  both  of  the  same  pitch,  and  in  the  first  instance,  of  the  same  thick- 
ness of  tooth — the  pitch  being  of  course  calculated  for  the  wooden  teeth  ; 
afterwards,  to  dress  down  the  teeth  of  the  iron  wheel  by  the  chipping-iwl, 
or  the  geer-cutting  machine  and  file,  to  the  exact  form  and  thickness,  as 
given  by  the  Table  ;  that  is,  to  a  thickness  in  relation  to  the  thickness  of 
the  wooden  teeth,  which  shall  have  the  ratio  of  25  to  38.  The  following 
table  may  be  useful  in  determining  the  relation  of  the  dimensions  of  the 
teeth  of  wheels  of  the  given  pitches,  and  the  power  which  they  are  capable 
of  transmitting  safely  at  the  various  speeds  named. 

To  find  the  power  which  a  wheel  is  capable  of  transmitting  for  other 
velocities  than  those  in  the  table : — For  6  feet  per  second,  double  the  result 
given  for  3  feet ;  for  8  feet,  double  the  result  at  4  feet,  and  so  on  ;  and  for 
lower  velocities  than  those  given,  divide  the  tabular  number  by  the  ratio 
which  they  bear  to  those  enumerated.  Thus,  for  2^  feet  velocity,  take 
half  the  result  at  5  feet,  and  so  of  other  velocities. 


Stress  in  Ibs. 

5  — 
Thickness  of 

Actual  pitches  to  which 

circle. 

lb>. 

inches. 

inches. 

400 

0.50 

H  to  H 

800 

O.T1 

1}    "    H 

1200 

O.ST 

H   "   2 

1600 

1.00 

2     «   2J 

2000 

1.12 

2i   »   2j 

2400 

1.22 

2*  "   2f 

2SOO 

1.32 

2*  "   2} 

3200 

1.41 

2i  "   3 

3600 

1.50 

8*   "   3} 

4000 

1.5S 

8t  "   8f 

4400 

1.66 

Sf  "   3} 

4800 

1.T3 

8*   "   34 

5200 

l.SO 

8J  "   3} 

5600 

1.8T 

3}   "   4 

6000 

1.94 

4     "   41 

168 


DRAWING    OF    MACHINERY. 


When  a  wheel  and  pinion,  which  differ  very  much  in  size,  work 
together,  the  teeth  of  the  latter,  on  account  of  their  unequal  thickness,  are 
capable  of  sustaining  much  less  pressure  than  the  teeth  of  the  wheel :  they 
are  in  effect,  if  not  in  fact,  much  reduced  in  thickness  ;  and  in  applying 


Pilch. 

Thickness 

Length 

Least 
breadth  of 

Velocity  of  the  wheel  at  the  pitch  circle. 

of  teeth. 

of  teeth. 

teeth. 

Three  feet 

Four  feet 

Fire  feet 

Seven  feet 

Eleven  feet 

per  second. 

per  second. 

per  second. 

per  second. 

per  second. 

Inches. 

Inches. 

Inches. 

Inches. 

H.   P. 

B.  P. 

H.  P. 

H    P 

B    P 

6 

2.9 

42 

8.4 

43.2 

57.6 

72. 

100.2 

158.4 

5* 

2.6 

3.S5 

7.7 

3G.3 

48.4 

60.5 

847 

133.1 

4 

1.9 

2.S 

5.6 

19. 

25.5 

32. 

45. 

70.5 

3* 

1.6 

2.45 

4.9 

1475 

19.5 

24.5 

3425 

54. 

3 

1.4 

2.1 

4.2 

11. 

14.5 

18. 

25. 

39.5 

2* 

1.2 

1.T5 

3.5 

7.5 

10. 

12.5 

17.5 

27.5 

2 

0.95 

1.4 

2.8 

4.75 

6.25 

8. 

11. 

17.25 

u 

O.S3 

1.225 

2.45 

3.5 

5. 

6.25 

8.5 

13.5 

4 

o.7i 

1.05 

2.1 

2.75 

3.5 

45 

6.25 

10. 

1} 

0.59 

0.875 

1.75 

2. 

2.5 

3.125 

42 

6.8 

l* 

0.53 

O.TS75 

1.575 

1.5 

2.25 

2.5 

3.5 

5.5 

i 

0.43 

0.7 

1.4 

1.2 

1.6 

2. 

2.S 

4.4 

1 

0.41 

0.6125 

1.225 

1. 

1.4 

1.75 

2.5 

3.S 

i 

0.36 

0.525 

1.05 

.7 

.9 

1.125 

1.5 

2.5 

1 

0.33 

0.4375 

0.675 

.5 

.625 

.75 

1. 

1.7 

i 

0.24 

0.35 

0.7 

.3 

.4 

.5 

.7 

1.1 

1 

0.18 

0.2625 

0.525 

.2 

.25 

.3 

.4 

.6 

i 

0.12 

0.175 

0.85 

.075 

.1 

.125 

.175 

.275 

rules  to  the  calculation  of  the  strength  of  wheels,  the  difference  of  size  of 
the  pair  ought  not  to  be  overlooked,  unless,  as  is  indeed  very  common  in 
practice,  the  deficiency  of  strength  be  made  up  to  the  pinion  by  a  flange 
cast  on  one  or  both  sides  of  the  rim,  of  the  same  depth  as  the  teeth,  and 
binding  these  together  like  the  staves  of  a  trundle.  In  this  case  the 
pinion  is  commonly  the  stronger  wheel  of  the  pair. 

In  the  construction  of  wheels,  the  problem  which  presents  itself  relative 
to  the  shape  of  the  teeth  is  this,  that  the  surfaces  of  mutual  contact  shall 
be  so  formed,  that  the  wheels  shall  be  made  to  turn  by  the  intervention  of 
the  teeth,  precisely  as  they  would  by  the  friction  of  their  circumferences. 

Fundamental  principle. — In  order  that  two  circles  A  and  B  (fig.  269) 
may  be  made  to  revolve  by  the  contact  of  the  surfaces  of  the  curves  in  m 
and  n  n  of  their  teeth  precisely  as  they,  would  by  the  friction  of  their  cir- 
cumferences, it  is  necessary  and  sufficient  that  a  line  drawn  from  the  point 
of  contact  t  of  the  teeth  to  the  point  of  contact  c  of  the  circumferences 
(pitch  circles),  should,  in  every  position  of  the  point  £,  be  perpendicular  to 
the  surfaces  of  contact  at  that  point ;  that  is,  in  the  language  of  mathe- 
maticians, that  the  straight  line  be  a  normal  to  both  the  curves  m  m  and 


DRAWING    OF   MACHINERY. 


169 


n  n.  The  principle  here  announced  exhibits  a  special  application  of  one 
particular  property  of  that  curve  known  to  mathematicians  as  the  epicy- 
cloid (see  page  76). 


C\l/0s> 


Of  epicycloidal  teeth. — The  simplest  illustration  of  the  action  of  epicy- 
cloidal  teeth  is  when  they  are  employed  to  drive  a  trundle,  as  represented 
in  fig.  270.  Let  it  be  assumed  that  the  staves  of  the  trundle  have  no  sen- 
sible thickness  ;  that  the  distance  of  their  centres  apart,  that  is  their  pitch, 
and  also  their  distance  from  the  centre  of  the  trundle,  that  is  their  pitch 
circle,  are  known.  The  pitch  circles  of  the  trundle  and  wheel  being  then 
drawn  from  their  respective  centres  B  and  A,  set  off  the  pitches  upon  these 
circumferences,  corresponding  to  the  number  of  teeth  in  the  wheel  and 
number  of  staves  in  the  trundle  ;  let  five  pins  a  5  c,  &c.,'  be  fixed  into  the 
pitch  circle  of  the  trundle  to  represent  the  staves,  and  let  a  series  of  epicy- 
cloidal arcs  be  traced  with  a  describing  circle,  equal  in  diameter  to  the 
radius  of  the  pitch  circle  of  the  trundle,  and  meeting  in  the  points  "klm n, 
&c.,  alternately  from  right  and  left.  If,  now,  motion  be  given  to  the  wheel 
in  the  direction  of  the  arrow,  then  the  curved  face  in  r  will  press  against 
the  pin  5,  and  move  it  in  the  same  direction  ;  but  as  the  motion  continues, 
the  pin  will  slide  upwards  till  it  reaches  m,  when  the  tooth  and  pin  will 
quit  contact.  Before  this  happens,  the  next  pin  a  will  have  come  into 
contact  with  the  face  a  I  of  the  next  tooth,  which  repeating  the  same 
action,  will  bring  the  succeeding  pair  into  contact ;  and  so  on  continually. 

To  allow  of  the  required  thickness  of  staves,  it  is  sufficient  to  diminish 
the  size  of  the  teeth  of  the  wheel  by  a  quantity  equal  to  the  radius  of  the 
staves  (sometimes  increased  by  a  certain  fraction  of  the  pitch  for  clear- 


170 


DRAWING    OF   MACHINERY. 


ance),  by  drawing  within  the  primary  epicycloids  at  the  required  distance 
another  series  of  curves  parallel  to  these.  In  practice,  a  portion  must  be 
cut  from  the  points  of  the  teeth,  and  also  a  space  must  be  cut  out  within 
the  pitch  circle  of  the  driver,  to  allow  the  staves  to  pass  ;  but  no  particular 
form  is  requisite,  the  condition  to  be  attended  to  is  simply  to  allow  of  suffi- 
cient space  for  the  staves  to  pass  without  contact. 

The  action  of  a  wheel  and  trundle  being  understood,  it  is  easy  to  com- 
prehend that  of  the  teeth  of  a  pair  of  wheels  of  the  ordinary  construction. 
Let  A.  and  B  of  fig.  271  be  respectively  the  centres  ot  a  wheel  and  pinion 

of  which  the  teeth  are  intended  to  be 
of  the  epicycloidal  form,  and  A  c  and 
B  c  their  primitive  radii.  To  lay  off 
the  teeth  of  this  pair,  having  deter- 
mined the  pitch  and  number  of  teeth 
in  the  wheel  and  pinion,  let  the  pitch 
lines  be  divided  into  as  many  equal 
parts,  setting  out  from  the  point  of 
contact  c,  as  there  are  teeth  in  them 
respectively.  Let  the  thickness  of 
the  teeth  be  next  set  off,  taking  c  a 
for  the  thickness  of  a  tooth  of  the 
wheel,  and  c  l>  for  that  of  a  tooth  of 
the  pinion.  Upon  the  radii  A  c  and 
B  c  as  diameters  describe  two  circles, 
having  also  their  point  of  contact  at 
Fis- 271-  c  and  their  centres  at  X  and  Y.  Xow 

let  the  circle  Y  be  made  to  roll  upon  the  pitch  line  of  the  wheel,  and  a 
point  in  its  circumference  at  c  will  describe  the  epicycloidal  arc  c  m,  and 
this  curve  determines  the  form  of  the  point  of  the  tooth  of  the  wheel.  In 
the  same  way  describe  the  epicycloidal  arc  c  n,  by  making  the  circle  X  to 
roll  upon  the  pitch  circle  of  the  pinion,  and  this  curve  will  determine  the 
form  of  the  part  of  the  tooth  of  the  pinion  beyond  the  pitch  line. 

The  curve  c  m  of  the  tooth  of  the  wheel  is  constantly  in  contact  with  the 
radius  B  c,  and  its  point  of  contact  is  at  the  same  time  situated  in  the  circum- 
ference of  the  circle  X  ;  the  contact  will  therefore  cease  when  the  extrem- 
ity m  of  the  tooth  becomes  the  point  of  contact ;  and  this  occurs  when  the 
point  m  has  arrived  at  the  circumference  of  Y.  If,  then,  an  arc  of  a  circle  be 
described  from  the  centre  A  with  the  radius  A  m,  its  pointy  of  intersection 
with  the  circumference  of  Y  is  that  at  which  the  tooth  ought  to  cease  to  act, 


DRAWING-    OF    MACHINERY.  171 

to  secure  uniformity  of  motion,  and  at  the  same  instant  that  another  new 
tooth  advances  into  geer.  The  determination  of  the  point/  limits  also  the 
useful  length  of  the  flank;  for  if  from  B  as  a  centre,  with  radius  B/,  an 
arc  be  described,  the  part  cf  of  the  radius  c  B  is  that  which  is  in  contact 
with  the  tooth  till  it  arrives  at  the  position  B/,  and  consequently  this  is 
the  useful  length  of  the  flank  of  the  pinion.  In  the  same  manner  it  may 
be  determined,  that  the  useful  length  of  the  flank  of  the  tooth  of  the  wheel 
c  a  is  the  portion  c  <j. 

To  find  the  form  of  the  portions  of  the  teeth  within  their  respective 
pitch  circles,  it  is  usually  considered  enough  that  sufficient  space  for  play 
be  allowed  whilst  the  tooth  remains  strong  enough  for  its  work.  As  every 
tooth  moves  between  two  flanks,  and  touches  only  one  of  them,  the  space 
may  be  bounded  literally  by  radial  lines  prolonged  towards  the  centre  of 
the  wheel.  The  bottom  of  the  space  being  sufficiently  removed  from  the 
pitch  circumference  to  allow  the  tops  of  the  teeth  of  the  pinion  to  pass 
round  without  touching  them,  may  be  described  by  arcs  of  a  circle  drawn 
from  the  centre.  In  practice,  and  especially  when  the  teeth  of  the  wheels 
are  small,  it  is  not  usually  considered  necessary  to  apply  strictly  the  form 
of  the  epicycloid  to  find  curves  of  the  teeth,  but  to  define  them  approxi- 
mately. 

The  forms  of  the  teeth  are  also  occasionally  described  by  arcs,  of  which 
the  radii  are  equal  to  the  pitch,  and  with  the  centres  taken  upon  the  pitch 
lines.  When  the  diameters  of  the  wheels  are  not  very  unequal,  and  the 
teeth  not  thick,  in  many  cases  the  curves  of  the  teeth  are  described  by  arcs 
drawn  from  the  centres  of  the  adjacent  teeth  upon  the  pitch  line.  This 
gives  a  radius  equal  to  the  pitch,  plus  half  the  thickness  of  a  tooth.  In 
like  manner,  the  sides  of  the  teeth  also  are  sometimes  described  by  arcs 
from  the  centres  of  the  adjacent  teeth,  giving  a  radius  equal  to  the  pitch, 
minus  half  the  thickness  of  a  tooth.  This  form  of  the  tooth  may  be  defec- 
tive, in  the  case  of  a  very  small  pinion  having  to  transmit  great  pressure, 
as  the  extremities  of  the  teeth  may  be  too  much  reduced.  In  this  case, 
the  curves  or  faces  of  the  teeth  may  be  described  with  radii  equal  to  three- 
fourths  of  the  pitch  ;  and  if  this  be  not  sufficient  curvature,  radii  equal  to 
some  smaller  fraction  of  the  pitch  may  be  used.  When,  on  the  contrary, 
the  pitch  is  large,  and  the  pressure  comparatively  small,  the  teeth  may  be 
too  short ;  this  will  be  remedied  by  employing  arcs,  of  which  the  radius  is 
one  and  a  half  or  twice  the  pitch.  In  practice,  the  ordinary  mode  in 
which  epicycloidal  teeth  are  set  out  is  by  the  mechanical  method  of  form- 
ing the  pattern  teeth  by  templets  (figs.  272,  273,  274,  275). 

Having  determined  the  pitch  of  the  teeth  and  the  radius  of  the  pitch 


172 


DRAWING   OF  MACHINERY. 


circle,  describe  on  a  thin  slip  of  wood — say  1  in.  thick — an  arc  of  the  pitch 
circle,  and  on  another  similar  slip  an  arc  of  a  circle  equal  to  the  diameter 


Fig.  272. 

of  the  wheel  to  the  points  of  the  teeth.  The  slips  are  then  cut  to  the  cir- 
cumferences of  the  circular  arcs  described  upon  them.  These  two  pieces 
so  prepared  are  fastened  together  by  screws,  as  in  the  fig.  272  ;  the  piece 


Fig.  273. 


A,  whose  edge  s  f  is  an  arc  of  the  pitch  circle,  is  fixed  upon  B,  whose 
edge  is  an  arc  of  the  extreme  circumference  of  the  wheel,  the  space  *  s  be- 
tween those  edges  being  in  breadth  equal  to  the  length  of  the  teeth  from 


Fig.  274. 


Fig.  275. 


the  pitch  circle  to  the  points.     This  done,  a  like  templet  is  prepared  for 
\\\Q  pinion  (fig.  273). 

The  pair  of  templets  being  thus  prepared,  two  tracing  points  m  m  are 


DRAWING   OF   MACHINERY.  173 

inserted  obliquely,  and  from  behind  into  the  piece  D  of  the  pinion  tem- 
plet. One  of  these  points  passes  out  at  the  edge  of  the  piece  C,  and  the 
other  at  the  edge  of  the  piece  D,  and  the  templets  are  then  placed  upon 
each  other,  as  shown  in  figs.  274  and  275  annexed,  so  that  the  circumfer- 
ence of  the  piece  C,  that  is  the  pitch  circumference  of  the  pinion,  shall 
meet  the  circumference  of  the  piece  A,  that  is  the  pitch  circumference  of 
the  wheel.  If  in  this  position  the  templets  be  made  to  roll  upon  each 
other  through  a  certain  arc,  pressing  them  at  the  same  time  slightly 
together,  the  tracing  points  will  mark  two  epicycloidal  curves  upon  the 
pieces  A  and  B,  as  v  and  r;  of  these  two  curves,  that  marked  -y,  which  is 
traced  on  the  face  of  the  piece  A,  will  be  the  curve  of  the  lower  portion  or 
flank,  and  that  marked  r  will  be  the  curve  of  the  upper  portion,  or  face  of 
a  tooth  of  the  wheel.  If  now  the  thickness  of  the  tooth  be  marked  off  on 
the  edge  of  the  piece  C,  that  is,  on  the  pitch  circle,  and  the  corresponding 
tracing  point  be  made  to  coincide  with  that  point,  the  curves  of  the  oppo- 
site side  of  the  tooth  will  be  formed  by  making  the  templets  roll  together 
in  the  contrary  direction.  A  complete  outline  of  a  tooth  of  the  wheel  is 
thus  described,  to  which  a  pattern  tooth  may  be  cut  and  used  to  shape  the 
teeth  by  in  making  the  wheel  pattern.  Instead  of  forming  pattern  teeth, 
many  prefer  to  lay  off  the  teeth  by  circular  arcs  coinciding  approximately 
with  the  epicycloidal  arcs  found  by  the  templets. 

The  preceding  mode  of  obtaining  the  curves  of  the  teeth  of  a  pair  of 
wheels  is  faulty  in  as  far  as  it  gives  a  form  of  teeth  smaller  at  the  root  than 
at  the  pitch  circles,  and  also  in  the  circumstance  that  a  pair  of  wheels, 
formed  in  the  manner  described,  will  only  work  correctly  with  each  other, 
and  not  with  wheels  of  any  other  numbers,  although  of  the  same  pitch. 
To  obviate  this,  it  is  necessary  to  employ  a  modification  of  the  ordinary 
practice. 

Let  a  thin  slip  of  wood  be  provided,  and  let  an  arc  of  the  pitch  circle 
be  struck  upon  it ;  divide  the  slip  into  two  portions  through  the  line  of 
this  arc  with  a  fine  saw  ;  one  part,  A,  will  have  a  concave,  and  the  other, 
B,  a  corresponding  convex  circular  edge.  Describe  an  arc  dd  of  the  pitch 
circle  upon  a  second  board  C  D,  upon  which  the  pattern  tooth  is  to  be 
drawn.  Fix  the  piece  B  upon  the  board,  so  that  its  circular  edge  may 
accurately  coincide  with  the  circumference  of  the  arc  d  d.  A  portion  of 
a  circular  plate  D  is  next  provided,  of  the  same  radius  which  it  is  proposed 
to  give  to  the  generating  circle  :  this  plate  has  a  fine  tracing  point  nip 
inserted  into  it,  and  projecting  slightly  from  its  under  surface,  and  accu- 
rately coinciding  with  its  circumference.  Having  set  off  the  ^hickness  of 
the  tooth  a  c  upon  the  pitch  circle  d  d,  so  that  twice  this  width  increased 


174 


DRAWING   OF   MACHINEET. 


Fig.  276. 


by  the  clearance  which  it  is  desired  the  teeth  should  have,  may  be  equal 
to  the  pitch,  the  generating  circle  D  is  made  to  roll  upon  the  convex  edge 

of  B ;  meantime  the  point  at  p  will 
trace  upon  the  board  the  curve  of 
the  faces  of  the  tooth,  having  caused 
the  point  to  coincide  successively 
with  the  two  points  a  and  c,  and 
the  circle  to  roll  from  right  to  left, 
and  vice  versa. 

Let  the  piece  B  be  now  re- 
moved, and  the  piece  A  applied 
and  fixed,  so  that  its  concave 
edge  may  accurately  coincide  with 
the  circular  arc  d  d;  then,  with 
the  same  circular  plate  D  pressed 
against  the  concave  edge  of  A,  and 
made  to  roll  upon  it,  the  point  at  p,  which  is  made  as  before  to  coincide 
successively  with  the  points  a  and  c,  will  trace  upon  the  surface  of  the 
board  C  D  the  two  hypocycloidal  arcs  a  5,  c  5,  which  form  the  flanks  of 
the  tooth.  The  complete  tooth,  thus  formed,  will  work  correctly  with  the 
teeth  similarly  described  iipon  any  other  wheel,  provided  the  pitch  of  the 
teeth  be  the  same,  and  the  same  generating  circle  D  be  used  to  strike  the 
curves  upon  the  two  wheels.  In  this  manner  the  general  forms  of  the 
teeth  of  the  pair  are  determined ;  and  it  only  remains  to  cut  them  off  at. 
such  lengths  that  they  shall  come  into  contact  in  the  act  of  passing  through 
the  line  of  centres. 

It  has  already  been  observed,  that  having  found  the  epicycloidal  curves 
of  the  teeth  by  means  of  the  templets,  a  common  method  of  proceeding  is 
to  find,  by  trial,  a  centre  and  small  radius,  by  which  the  arc  of  a  circle 
can  be  described  that  will  coincide  nearly  with  the  templet-traced  curve. 
A  more  commodious  and  certain  method  of  determining  the  centre  and 
radius  of  the  approximate  arc  has  been  supplied  by  Prof.  "Willis  in  the 
construction  of  his  Odontograpli,  manufactured  by  Messrs.  Holtrapfel  & 
Co.,  London,  for  description  of  which  see  Appletojis*  Dictionary  of  Me- 
chanics, pp.  819,  820. 

Involute  teeth. — Involute  teeth  have  the  disadvantage  of  being,  when 
in  contact,  too  much  inclined  to  the  radius,  by  which  an  undue  pressure  is 
transferred  to  their  axes.  Their  mutual  friction  is  thereby  little  affected, 
but  that  of  the  axes  is  increased,  and  their  journals  are  more  speedily 
worn.  But  they  have  at  the  same  time  the  advantage  of  working  with 


DRAWING    OF    MACHINERY. 


175 


more  accuracy  under  derangement  and  incorrectness  of  fitting,  and  any 
pairs  of  them  will  work  truly  together  in  sets  within  certain  limits,  how- 
ever different  in  diameters,  the  pitch  being  the  same. 

To  describe  this  curve  for  the  teeth  of  a  pair,  of  which  the  radii  of  the 
pitch  circles  and  pitch  of  the  tooth  are  determined,  we  may  employ  the 
mode  illustrated  by  fig.  277.  Let  A  and  B  be  the  centres  of  the  pair,  and 
e  1)  be  their  pitch  lines ;  join  A  and  B  by  a  right  line  passing  through  c; 
from  this  last  point  draw  c  d,  c  d,  perpendicular  to  the  radials  B  d,  A  d, 
and  cutting  them  in  d  and  d;  this  line  d  d  is  then  a  common  normal  to 
the  teeth  in  contact,  and  the  perpendiculars  A  d,  B  d,  are  the  radii  of  the 
involute  circles  which  form  the  acting  faces  of  the  teeth. 


A.' 

Fig.  277. 


u 

Fig.  278. 


Having  thus  elucidated  the  principles  of  the  operation  of  toothed  geer- 
ing,  and  the  form  and  proportions  to  be  given  to  their  parts,  we  proceed 
now  to  give  complete  drawings  of  different  toothed  wheels ;  premising, 
that  except  in  case  of  working  drawings  to  full  size,  arcs  of  circles  are 
almost  invariably  used  by  the  draughtsman  to  describe  the  forms  of  the 
teeth  ;  and  that  in  practice,  too  little  attention  is  paid  to  the  construction 
of  epicycloidal  and  involute  teeth.  In  many  drawings  details  are  unneces- 
sary, and  spur  geers  are  represented  by  pulleys  and  bevel  geers  as  in  fig. 
278,  the  pitch  being  written  in  in  figures. 


PROJECTIONS    OF    A    SPUR    WHEEL. 

PL  XVII. — To  draw  side  elevation  (fig.  1),  an  edge  view  (fig.  2),  and  a 
vertical  section  (fig.  3),  of  a  spur  wheel  with  34  teeth,  and  a  pitch  of  two 
inches  : 

Determine  the  radius  of  the  pitch  circle  from  table,  page  161 ;  against 
2  in.  we  find  in  the  second  column  .6366  ;  .6366  X  54  =  34.38,  the  diam- 
eter. Draw  the  central  line  A  C  B  and  the  perpendicular  D  E ;  on  C  as  a 
centre,  with  a  radius  17.19,  describe  the  pitch  circle,  and  divide  it  into  54 


176  DRAWING    OF    MACHINERY. 

equal  parts.  To'  effect  this  division,  without  fraying  by  repeated  trials 
that  part  of  the  paper  on  which  the  teeth  are  to  be  represented,  describe 
from  the  same  centre  C',  with  any  convenient  radius,  a  circle  a  I  c  d;  with 
the  same  radius  divide  its  circumference  into  six  equal  parts,  and  sub- 
divide each  sixth  into  nine  equal  parts,  and  draw  radii  to  the  centre  C' ; 
these  radii  will  cut  the  pitch  circle  at  the  required  number  of  points. 
Divide  the  pitch  (2  in.)  into  15  equal  parts ;  mark  off  beyond  the  pitch 
circle  a  distance  equal  to  5i  of  these  parts,  and  within  it  a  distance  equal 
to  6£  parts  (see  page  163),  and  from  the  centre  C  describe  circles  passing 
through  these  points  ;  these  circles  are  projections  of  the  cylinders  bound- 
ing the  points  of  the  teeth  and  the  roots  of  the  spaces  respectively. 

In  forming  the  outlines  of  the  teeth,  the  radii  which,  by  their  intersec- 
tions with  the  pitch  circle,  divide  it  into  the  required  number  of  parts, 
may  be  taken  as  the  centre  lines  of  each  tooth.  The  thickness  of  the  tooth, 
measured  on  the  pitch  circle,  is  T7j  of  the  pitch,  and  the  width  of  the 
space  is  equal  to  T8j.  These  distances  being  set  off,  take  in  the  com- 
passes the  length  of  the  pitch,  and  from  the  centre  y  describe  a  circular 
arc  h  ij  and  from  the  centre^',  with  the  same  radius,  describe  another  arc 
Ti  If,  touching  the  former  ;  these  arcs  being  terminated  at  the  circles  bound- 
ing the  points  of  the  teeth  and  the  bottoms  of  the  spaces  respectively,  form 
the  curve  of  one  side  of  a  tooth.  The  other  side  is  formed  in  a  similar 
manner,  by  drawing  from  the  centre  I  the  arc  in  n,  and  from  the  centre  p 
the  arc  m  0,  and  so  on  for  all  the  rest  of  the  teeth. 

The  teeth  having  been  thus  completed,  we  proceed  to  the  delineation 
of  the  rim,  arms,  and  eye  of  the  wheel.  The  thickness  of  the  rim  is  usu- 
ally made  equal  to  that  of  the  teeth,  namely,  y7^  of  the  pitch,  which  dis- 
tance is  accordingly  set  off  on  a  radius  within  the  circle  of  the  bottoms  of 
the  spaces,  and  a  circle  is  described  from  the  centre  C  through  the  point 
q  thus  obtained.  AVithin  the  rim,  a  strengthening  feather  q  r,  in  depth 
about  £  of  the  thickness  of  the  rim,  is  generally  formed,  as  shown  in 
the  plate.  The  eye,  or  central  aperture  for  the  reception  of  the  shaft,  is 
then  drawn  to  the  specified  diameter,  as  also  the  circle  representing  the 
thickness  of  metal  round  the  eye,  which  is  usually  made  equal  to  the  pitch 
of  the  wheel. 

To  draw  the  arms,  from  the  centre  C,  with  the  radius  C  u  equal  to  the 
pitch,  describe  a  circle  ;  draw  all  the  radii,  as  C  L,  which  are  to  form  the 
centre  lines  of  the  arms,  and  set  off  the  distance  L  v,  equal  to  f  pitch,  on 
each  side  of  these  radii  at  the  inner  circumference  of  the  rim,  and  through 
all  the  points  thus  obtained  draw  tangents  to  the  circle  passing  through 
u.  The  contiguous  arms  are  rounded  off  into  each  other  by  arcs  of  circles, 


DRAWING    OF   MACHINERY.  177 

whoso  centres  are  obtained  by  the  following  construction : — Taking,  for 
example,  the  arc  M  P  Q,  it  is  obvious  that  its  centre  is  situated  in  the 
straight  line  0  E  which  divides  equally  the  interval  between  two  contigu- 
ous arms.  Having  fixed  the  point  P  (which  should  be  at  the  same  distance 
from  t  as  the  breadth  of  the  feather  at  the  back  of  the  rim)  draw  through 
it  a  perpendicular  R  P  to  the  line  C  E  ;  the  question  now  becomes  simply 
a  geometrical  problem,  to  draw  a  circle  touching  the  three  straight  lines 
M  X,  P  R,  and  S  Q.  Divide  the  angle  P  R  M  into  two  equal  parts  by 
the  straight  line  R  O,  which  cuts  C  E  in  the  point  O,  the  centre  of  the 
circle  required  ;  its  radius  is  the  line  O  M  perpendicular  to  M  1ST.  If,  now, 
a  cirple  be  drawn  from  the  centre  C,  Avith  the  radius  C  O,  its  intersection 
with  the  radii  bisecting  all  the  intervals  between  the  arms  will  give  the 
remaining  centres,  such  as  O',  of  the  arcs  required ;  and  the  circle  pass- 
ing similarly  through  M,  marks  all  the  points  of  contact  M  Q  M',  &c.  To 
draw  the  small  arcs  terminating  the  extremities  of  the  arms,  set  off  upon 
the  line  C  E,  within  the  point  r,  the  required  radius  of  the  arcs,  and  from 
the  centre  C  with  a  radius  C  w  describe  a  circle  ;  the  distance  r  w  being 
then  transferred  to  the  extremities  of  the  arms  at  the  points  where  they 
are  cut  by  the  circle,  as  at  S  a?,  will  give  the  centres  of  the  arcs  required. 
Draw  the  central  web  of  the  arm  by  lines  parallel  to  their  radii,  making 
the  thickness  about  f  inch  for  wheel  of  about  this  size. 

Having  thus  completed  elevation,  the  construction  of  the  edge  view 
and  vertical  section  becomes  comparatively  simple.  Draw  the  perpendicu- 
lars F  G  and  II I  (figs.  2, 3)  as  central  lines  in  the  representations ;  set  off  on 
each  side  of  these  lines  half  the  breadth  .of  the  teeth,  and  draw  parallels  ; 
project  the  teeth  of  fig.  1  upon  fig.  2,  by  drawing  through  all  the  visible 
angular  points  straight  lines  parallel  to  A  B,  and  terminated  at  either  ex- 
tremity by  the  verticals  representing  the  outlines  of  the  breadth  of  the 
wheel ;  project  in  like  manner  the  circles  of  the  hub  ;  lay  off  half  length 
on  each  side  of  F  G,  and  draw  parallels  to  it.  The  section  (fig.  3)  is  sup- 
posed to  be  made  on  the  line  D  E  of  the  elevation  ;  project,  as  in  fig.  2, 
those  portions  which  will  be  visible  in  this  section,  and  shade  those  parts 
which  are  in  section.  The  arms  are  made  tapering  in  width,  and  somewhat 
less  than  the  face  of  the  wheel. 

Since  the  two  projections  (figs.  1  and  3)  are  not  sufficient  to  exhibit  fully 
the  true  form,  a  cross  section  of  one  of  them  is  given  at  fig.  4 ;  this  section 
is  supposed  to  be  made  by  a  plane  passing  through  X  X'  and  Y  Y'.  The 
points  y,  2,  in  fig.  1,  and  corresponding  lines  in  fig.  3,  represent  the  edges 
of  key-seat. 

12 


ITS  DRAWING   OF   MACHINERY. 


OBLIQUE   PROJECTION    OF   A    SPUR   WHEEL. 

Plate  XIX. — In  drawing  a  spur  wheel  or  other  object  in  an  oblique 
position  with  respect  to  the  vertical  plane  of  projection,  it- is  necessary,  in 
the  first  place,  to  lay  down  the  elevation  and  plan  as  if  it  were  parallel  to 
that  plane,  as  represented  at  figs.  1  and  2.  Then  transfer  the  plan  to  fig. 
4,  giving  it  the  same  inclination  with  the  ground  line  which  the  wheel 
ought  to  have  in  relation  to  the  vertical  plane  ;  and  assuming  that  the 
horizontal  line  A  B  represents  the  axis  of  the  wheel,  both  in  the  parallel 
and  oblique  positions,  the  centre  of  its  front  face  in  the  latter  position  will 
be  determined  by  the  intersection  of  a  perpendicular  raised  from  the  point 
C'  (fig.  4)  with  that  axis.  Now  it  is  obvious,  that  if  we  take  any  point,  as 
a  in  fig.  1,  the  projection  of  that  point  on  fig.  3  must  be  in  the  line  a  #, 
parallel  to  A  B ;  and  further,  this  point  being  projected  at  a'  (fig.  4),  it 
must  be  in  the  perpendicular  a'  a;  therefore  the  intersection  of  these  two 
lines  is  the  point  required.  Thus  all  the  remaining  points  J,  c1,  cl,  &c.,  may 
be  obtained  by  the  intersections  of  the  perpendiculars  raised  from  the 
points  &',  c',  d',  &c.  (fig.  4),  respectively,  with  the  horizontals  drawn 
through  the  corresponding  points  in  fig.  1.  It  will  also  be  observed,  that 
since  the  points  e  and/,  in  the  further  face  of  the  wheel,  have  their  projec- 
tions in  a  and  5  (fig.  1),  their  oblique  projections  will  be  situated  in  the 
lines  a  a  and  5  5,  but  they  are  also  at  e  and//  consequently,  the  lines  e  a 
and/  b  are  the  oblique  projections  of  the  edges  a'  e'  and  It' '/'.  We  have 
now  to  remark,  that  all  the  circles  which,  in  the  rectangular  elevation 
(fig.  1),  have  been  employed  in  the  construction  of  this  w^heel,  are  pro- 
jected in  the  oblique  view  into  ellipses,  the  length  and  position  of  whose 
axes  may  be  determined  without  any  difiiculty  ;  lor  since  the  plane  F'  G', 
in  which  these  circles  are  situated,  is  vertical,  the  major  axes  of  all  the 
ellipses  in  question  will  obviously  be  perpendicular  to  the  line  A  B,  and 
equal  to  the  diameters  of  the  circles  of  which  they  are  respectively  the  pro- 
jections ;  and  the  minor  axes,  representing  the  horizontal  diameters,  will 
all  coincide  with  the  line  A  B.  Thus,  to  obtain  the  ellipse  into  which  the 
pitch  circle  is  projected,  it  is  only  necessary  to  set  off  upon  the  vertical 
D  E  (fig.  3),  above  and  below  the  point  C,  the  radius  of  the  pitch-circle, 
whose  horizontal  diameter  ij  being  at  i'j'  (fig.  4)  is  projected  to  ij  (fig.  3) ; 
and  thus  having  obtained  the  major  and  minor  axes,  the  ellipse  in  question 
may  easily  be  constructed.  The  intersection  of  the  horizontal  lines  gg,  h  k, 
&c.,  with  this  circle  gives  the  thickness  of  the  teeth  at  the  pitch  line ;  and 
by  projecting  in  the  same  manner  the  circles  bounding  the  extremities  and 


DRAWING    OF   MACHINERY.  179 

roots  of  the  teeth,  these  points  in  each  individual  tooth  may  be  determined 
by  a  similar  process.  But  since,  in  cases  where  strict  accuracy  is  required, 
a  greater  number  of  points  is  necessary  for  the  construction  of  the  curva- 
ture of  the  teeth,  two  additional  circles  m  n  and  o  p  may  be  drawn  on  fig. 
1,  and  projected  to  fig.  3,  and  the  points  of  their  intersection  with  the 
curves  of  the  teeth  projected  to  fig.  3,  where  the  corresponding  points  are 
indicated  by  the  same  letters. 

It  is  almost  unnecessary  to  observe,  that  the  instructions  we  have  given 
for  the  drawing  of  the  anterior  face  F'  G'  of  the  wheel  are  equally  appli- 
cable to  the  posterior  H'  I',  which  is  parallel  to  it,  and  in  all  respects  the 
same  ;  the  common  centre  of  all  the  circles  in  it  being  at  O'  (fig.  4),  is  pro- 
jected to  O  in  fig.  3.  Hence,  it  will  be  easy  to  construct  the  ellipses 
representing  these  circles  in  the  oblique  projection,  and  consequently  to 
determine  the  points  e,  /,  £',  &c.,  in  the  curvature  of  the  teeth  ;  observing, 
that  as  their  centre  lines  converge  to  C  in  the  front  face,  they  all  tend  to 
O  in  the  remoter  surface,  which  is,  however,  for  the  most  part  concealed 
by  the  former. 

It  would  be  superfluous  to  enter  into  any  details  regarding  the  con- 
struction of  the  oblique  view  of  the  rim,  eye,  and  arms  which  are  drawn 
upon  precisely  similar  principles  to  those  we  have  already  so  fully  ex- 
plained. 

PROJECTIONS    OF    A    BEVIL    WHEEL. 

Plate  XXI. — Fig.  1  is  a  face  view,  fig.  2  an  edge  view,  and  fig.  3  a 
vertical  transverse  section.  We  have  explained  (page  159)  the  determina- 
tion of  the  division  of  the  angle  of  inclination  of  the  axes  of  a  pair  of  bevil 
wheels ;  their  size  and  proportion  are  to  be  determined  by  the  rules  given 
for  spur-wheels ;  thus,  consider  the  base  of  the  cone  A  B  (figs.  2,  3)  as  the 
diameter  of  the  pitch  circle  of  a  spur-wheel,  and  proportion  the  pitch,  form, 
and  breadth  of  teeth,  according  to  the  stress  to  which  they  are  to  be  subjected. 

Having  determined  and  laid  down,  according  to  the  required  condi- 
tions, the  axis  O  S  of  the  primitive  cone,  the  diameter  A  B  of  its  base,  the 
angle  A  S  O  which  the  side  of  the  cone  makes  with  the  axis,  and  the 
straight  lines  A  o,  D  o',  perpendicular  to  A  S,  and  representing  the  sides 
of  two  cones,  between  which  the  breadth  of  the  wheel  (or  length  of  the 
teeth)  is  comprised,  the  first  operation  is  to  divide  the  primitive  circle,  de- 
scribed with  the  radius  A  C,  into  a  number  of  equal  parts  corresponding 
to  the  number  of  teeth  or  pitch  of  the  wheel.  Then  upon  the  section  (fig. 
3),  draw  with  the  radius  o  A  or  o  B,  supposed  to  move  parallel  to  itself, 


180  DRAWING   OF   MACHINERY. 

outside  the  figure  a  small  portion  of  a  circle,  upon  which  construct  the 
outlines  of  a  tooth  M,  and  of  the  rim  of  the  wheel,  with  the  same  propor- 
tions and  after  the  same  manner  as  we  have  explained  in  reference  to 
spur-wheels  ;  set  off  from  A  and  B  the  points  a,  d,  andy,  denoting  respec- 
tively the  distances  from  the  pitch  line  to  the  points  and  roots  of  the  teeth, 
and  to  the.  inside  of  the  rim,  and  join  these  points  to  the  vertex  S  of  the 
primitive  cone,  terminating  the  lines  of  junction  at  the  lines  D  </,  E  o' '; 
the  figure  a  1)  c  d  will  represent  the  lateral  form  of  a  tooth,  and  the  figure 
c  df  c,  a  section  of  the  rim  of  the  wheel,  by  the  aid  of  which  the  face  view 
(fig.  1)  may  easily  be  constructed. 

The  points  a,  5, •<?,  c?,  and  <?,  having  been  projected  upon  the  vertical 
diameter  A7  Br,  describe  from  the  centre  C'  a  series  of  circles  passing 
through  the  points  thus  obtained,  and  draw  any  radius,  as  C'  L,  passing 
through  the  centre  of  a  tooth.  On  either  side  of  the  point  L  set  off  the 
distances  L  k,  L  /,  making  up  the  thickness  of  the  tooth  M  at  the  point, 
and  indicate,  in  like  manner,  upon  the  circles  passing  through  the  points 
B7  and  d',  its  thickness  at  the  pitch  line  and  root ;  then  draw  radii  through 
the  points  i,  I,  &,  g,  m,  &c.,  terminating  them  respectively  at  the  circles 
forming  the  projections  of  the  corresponding  parts  at  the  inner  extremity 
of  the  teeth ;  these  radial  lines  will  represent  the  rectilinear  edges  of  all 
the  teeth.  The  curvilinear  outlines  may  be  delineated  by  arcs  of  circles, 
tangents  to  the  radii  g  C'  and  i  C',  and  passing  through  the  points  obtained 
by  the  intersections  of  the  radii  and  the  various  concentric  circles.  The 
radii  of  these  circular  arcs  may  in  general,  as  in  the  case  of  spur  wheels, 
be  taken  equal  to  the  pitch,  and  their  centres  upon  the  interior  and  exte- 
rior pitch-circles ;  thus  the  points  g  and  i,  n  and  <?,  for  example,  are  the 
centres  foi:  the  arcs  passing  through  the  corresponding  points  in  the  next 
adjacent  teeth,  and  vice  versa. 

The  drawing  of  the  teeth  in  the  edge  view  (fig.  2),  and  of  such  portions 
of  them  as  are  visible  in  the  section  (fig.  3),  is  sufficiently  explained  by  in- 
spection of  the  lines  of  projection  which  we  have  partially  introduced 
into  the  plate  for  this  purpose.  We  have  only  to  remark,  that  in  the  con- 
struction of  these  views,  every  point  in  the  principal  figure  from  which 
they  are  derived  is  situated  upon  the  projection  of  the  circle  drawn  from 
the  centre  C',  and  passing  through  that  point.  Thus  the  points  g  and  «', 
for  example,  situated  upon  the  exterior  pitch-circle,  will  be  determined  in 
fig.  2  by  the  intersection  of  their  lines  of  projection  with  the  base  A  B  of 
the  primitive  cone  ;  and  the  points  k  and  I  will  be  upon  the  straight  line 
passing  through  a  a  (fig.  3),  and  so  on.  Farther,  as  the  lateral  edges  of 
air  the  teeth  in  fig.  1  are  radii  of  circles  drawn  from  the  centre  C',  so  in 


DRAWING    OF   MACHINERY. 


181 


fig.  2  they  are  represented  by  lines  drawn  through  the  various  points 
found  as  above  for  the  outer  extremities  of  the  teeth,  and  converging 
towards  the  common  apex  S  ;  while  the  centre-lines  of  the  exterior  and 
interior  extremities  themselves  all  tend  to  the  points  o  and  o'  respectively. 
This  circumstance  will  suggest  a  mode  of  materially  simplifying  the  opera- 
tion of  drawing  the  edge  view  of  the  teeth  when  the  wheels  are  small,  or 
executed  to  a  small  scale  ;  and  in  all  cases  it  affords  a  means  of  testing 
the  accuracy  of  the  operations,  if  the  method  of  projecting  numerous  points 
be  adopted. 

Skew-bevels. — When  the  axes  of  wheels  are  inclined  to  each  other,  and 
yet  do  not  meet  in  direction,  and  it  is  proposed  to  connect  them  by  a 
single  pair  of  bevels,  the  teeth  must  be  inclined  to  the  base  of  the  frusta  to 


Fig.  279. 


allow  them  to  come  into  contact.  Set  off  a  e  equal  to  the  shortest  distance 
between  the  axes,  (called  the  eccentricity^)  and  divide  it  in  c,  so  that  a  c 
is  to  e  c  as  the  mean  radius  of  the  frustum  to  the  mean  radius  of  that  with 
which  it  is  to  work ;  draw  c  in  d  perpendicular  to  a  e.  The  line  c  m  d  gives 


182  DRAWING   OF   MACHINERY. 

the  direction  of  the  teeth ;  and  if  from  the  centre  a,  with  radius  a  c,  a 
circle  be  described,  the  direction  of  any  tooth  of  the  wheel  will  be  a  tan- 
gent to  it,  as  at  c.  Draw  the  line  d  e  perpendicular  to  c  m  d,  and  with  a 
radius  d  e  equal  to  c  e  describe  a  circle  ;  the  direction  of  the  teeth  of  the 
second  wheel  will  be  tangents  to  this  last,  as  at  d. 


SYSTEM    COMPOSED    OF   A   PINION   DRIVING    A   RACK. 

PL  XXIII.,  fig.  1.— The  pitch  line  M  N  of  the  rack,  and  the  primitive 
circle  A  B  D  of  the  pinion  being  laid  down  touching  one  another,  divide 
the  latter  into  twice  the  number  of  equal  parts  that  it  is  to  have  of  teeth, 
and  set  off  the  common  distance  of  these  parts  upon  the  line  M  K,  as  many 
times  as  may  be  required ;  this  marks  the  thickness  of  the  teeth  and  width 
of  the  spaces  in  the  rack.  Perpendiculars  drawn  through  all  these  points 
to  the  solid  part  of  the  rack  will  represent  the  flanks  of  the  teeth  upon 
which  those  of  the  pinion  are  to  be  developed  in  succession.  The  curvature 
of  these  latter  should  be  an  involute  A  c  of  the  circle  A  B  D.  The  teeth 
might  be  cut  off  at  the  point  of  contact  d  upon  the  line  M  "N,  for  at  this 
position  the  tooth  A  begins  its  action  upon  that  of  the  rack  E ;  but  it  is 
better  to  allow  a  little  more  length  ;  in  other  words,  to  describe  the  circle 
bounding  the  points  of  the  teeth  with  a  radius  somewhat  greater  than  C  d. 

"With  regard  to  the  form  of  the  spaces  in  the  rack,  all  that  is  required 
is  to  set  off  from  M  Is",  as  at  the  point  e,  a  distance  slightly  greater  than 
the  difference  A  a  of  the  radius  of  the  pitch  circle,  and  that  of  the  circle 
limiting  the  points  of  the  teeth,  and  through  this  point  to  draw  a  straight 
line  F  G  parallel  to  M  K  From  this  line  the  flanks  of  all  the  teeth  of  the 
rack  spring,  and  their  points  are  terminated  by  a  portion  of  a  cycloid  A  5, 
which,  however,  may  in  most  instances  be  replaced  by  an  arc  of  a  circle. 
The  depth  of  the  spaces  in  the  pinion  obviously  depends  upon  the  height 
of  this  curved  portion  of  the  teeth ;  their  outline  is  formed  by  a  circle 
drawn  from  the  centre  C,  with  a  radius  a  little  less  than  the  distance  from 
this  point  to  the  straight  line,  bounding  the  upper  surface  of  the  teeth  of 
the  rack. 


SYSTEM  COMPOSED    OF   A   RACK   DRIVING   A    PINION. 

In  this  case  the  construction  is  in  all  respects  identical  with  that  of  the 
preceding  example,  with  this  exception,  that  the  form  proper  to  be  given 
to  the  teeth  of  the  rack  is  a  cycloid  generated  by  a  point  A  in  the  circum- 


DRAWING    OF   MACHINERY.  183 

ference  of  the  circle  A  E  C,  in  rolling  on  the  line  M  1ST.     The  curvature 
of  the  teeth  of  the  pinion  is  an  involute  as  before. 


SYSTEM  COMPOSED    OF   A   WHEEL    AND   TANGENT,    OB   ENDLESS    SCREW. 

Fig.  2. — In  the  construction  of  this  variety  of  geering,  we  must  first  fix 
upon  the  number  of  teeth  in  the  wheel,  and  the  distance  of  its  centre  from 
the  axis  of  the  screw.  Then  conceive  a  plane  passing  through  the  axis 
E  F  of  the  screw,  parallel  to  the  face  of  the  wheel,  and  let  C  be. the  centre 
of  its  primitive  circle.  If  now  a  perpendicular  C  G  be  drawn  from  C  upon 
E  F,  and  C  A  be  taken  as  the  radius  of  the  pitch  circle  B  A  D  of  the 
wheel,  the  difference  A  G  will  represent  the  radius  of  a  cylinder,  which 
may  be  termed  the  primitive  cylinder  of  the  screw ;  and  a  line  M  N  drawn 
through  A,  parallel  to  E  F,  will  be  a  generatrix  of  that  cylinder,  which 
will  serve  the  purpose  of  determining  the  form  of  the  teeth. 

The  section  having  been  made  through  the  axis,  the  question  obviously 
resolves  itself  into  the  case  of  a  rack  driving  a  pinion  ;  consequently  the 
curve  of  the  teeth,  or  rather  thread,  of  the  screw  should  be  simply  a  cycloid 
generated  by  a  point  in  the  circle  A  E  C,  described  upon  A  C  as  a  diam- 
eter, and  rolling  upon  the  straight  line  M  1ST.  It  is  to  be  remarked,  fur- 
ther, that  the  outlines  of  the  teeth  are  helical  surfaces  described  about  the 
cylinder  forming  the  screw,  with  the  pitch  A  5  equal  to  the  distance,  mea- 
sured upon  the  primitive  scale,  between  the  corresponding  points  of  two 
contiguous  teeth.  These  curves  have  been  drawn  on  our  figure,  but  being 
for  the  most  part  concealed,  they  are  expressed  by  dotted  lines.  The  teeth 
of  the  wheel  are  not,  as  in  ordinary  kinds  of  geering,  set  perpendicularly 
to  the  plane  of  its  face,  but  at  an  angle,  and  with  surfaces  corresponding 
to  the  inclination  and  helical  form  of  the  thread  of  the  screw.  In  some 
instances,  the  points  of  the  teeth  and  bottoms  of  the  spaces  are  formed  of  a 
concave  outline  adapted  to  the  convexity  of  the  screw,  in  order  to  present 
as  much  bearing  surface  as  possible  to  its  action.  In  this  kind  of  geering, 
for  obvious  reasons,  it  is  invariably  the  screw  that  imparts  the  motion. 

Fig.  3  represents  an  edge  elevation  of  the  wheel,  projected  as  in  pre- 
vious examples. 

SYSTEM    COMPOSED    OF   AN    INTERNAL    SPUR-WHEEL    DRIVING    A   PINION. 

PI.  XXY.,  fig.  1.— The  form  of  the  teeth  of  the  driving  wheel  is  in  this 
instance  determined  by  the  epicycloid  described  by  a  point  in  the  circle 
A  E  C,  rolling  on  the  concave  circumference  of  the  primitive  circle  M  A  N". 


184  .  DRAWING   OF   MACHINERY. 

The  points  of  the  teeth  are  to  be  cut  off  by  a  circle  drawn  from  the  centre 
of  the  internal  wheel,  and  passing  through  the  point  E,  which  is  indi- 
cated, as  before,  by  the  contact  of  the  curve  with  the  flank  of  the  driven 
tooth. 

The  wheel  being  supposed  to  be  invariably  the  driver,  the  curved  por- 
tion of  the  teeth  of  the  pinion  may  be  very  small.  This  curvature  is  a 
part  of  an  epicycloid  generated  by  a  point  in  the  circle  MAX  rolling 
upon  BAD. 


SYSTEM   COMPOSED    OF   AX   INTERNAL   WHEEL   DRIVEN   BY   A   PINION. 

Fig.  2. — This  problem  involves  a  circumstance  which  has  not  hitherto 
come  under  consideration,  and  which  demands,  consequently,  a  different 
mode  of  treatment  from  that  employed  in  the  preceding  cases.  The  epicy- 
cloidal  curve  A  #,  generated  by  a  point  in  the  circle  having  the  diameter 
A  O,  the  radius  of  the  circle  MAX,  and  which  rolls  upon  the  circle 
BAD,  cannot  be  developed  upon  the  flank  A  &,  the  line  described  by 
the  same  point  in  the  same  circle  in  rolling  upon  the  concave  circumfer- 
ence MAX;  and  for  this  obvious  reason,  that  that  curve  is  situated  with- 
out the  circle  BAD,  while  the  flank,  on  the  contrary,  is  within  it.  It 
becomes  necessary,  therefore,  in  order  that  the  pinion  may  drive  the  wheel 
uniformly  according  to  the  required  conditions,  to  form  the  teeth  so  that 
they  shall  act  always  iipon  one  single  point  in  those  of  the  wheel.  This 
may  be  most  advantageously  effected  by  taking  for  the  curvature  of  the 
teeth  of  the  pinion  the  epicycloid  A  d  described  by  the  point  A  in  the 
circle  M  A  X,  rolling  over  the  circle  BAD.  It  will  be  observed  that,  as 
in  the  preceding  examples,  the  tooth  E  of  the  pinion  begins  its  action  upon 
the  tooth  F  of  the  wheel  at  the  point  of  contact  of  their  respective  primi- 
tive circles,  and  that  it  is  unnecessary  that  it  should  be  continued  beyond 
the  point  £,  because  the  succeeding  tooth  II  will  then  have  been  brought 
into  action  upon  G ;  consequently  the  teeth  of  the  wheel  might  be  bounded 
by  a  circle  passing  through  the  point  c.  It  is,  however,  one  of  the  prac- 
tical advantages  which  this  species  of  geering  has  over  wheels  working  ex- 
ternally, that  the  surfaces  of  contact  of  the  wheel  and  pinion  admit  of 
being  more  easily  increased  ;  and  by  making  the  teeth  somewhat  longer 
than  simple  necessity  demands,  the  strain  may  be  diffused  over  two  or 
more  teeth  at  the  same  time.  The  flanks  of  the  teeth  of  the  wheel  are 
formed  by  radii  drawn  to  the  centre  O,  and  their  points  are  rounded  off 
to  enable  them  to  enter  freely  into  the  spaces  of  the  pinion. 


DRAWING    OF   MACHINERY.  185 


PROJECTIONS     OF     ECCENTRICS. 

The  term  eccentric  is  applied  in  general  to  all  such  curves  as  are  com- 
posed of  points  situated  at  unequal  distances  from  a  central  point  or  axis. 
The  ellipse,  the  curve  called  the  heart,  and  even  the  circle  itself,  when 
supposed  to  be  fixed  upon  an  axis  which  does  not  pass  through  its  centre, 
are  examples  of  eccentric  curves. 

The  object  of  such  curves,  which  are  of  frequent  occurrence  in  machin- 
ery, is  to  convert  a  rotatory  into  an  alternating  rectilinear  motion  ;  and 
their  forms  admit  of  an  infinite  variety,  according  to  the  nature  of  the 
motion  desired  to  be  imparted.  Examples  of  their  application  occur  in 
many  arrangements  of  pumps,  presses,  valves  of  steam-engines,  spinning 
and  weaving  machines,  &c. 

Fig.  1,  pi.  XXYII. — To  draw  the  eccentrical  symmetrical  curve  called 
the  heart,  which  is  such  as,  when  revolving  with  a  uniform  motion  on  its 
axis,  to  communicate  to  a  movable  point  A,a  uniform  rectilinear  motion  of 
ascent  and  descent. 

Let  C  be  the  axis  or  centre  of  rotation  upon  which  the  eccentric  is 
fixed,  ami  which  is  supposed  to  revolve  uniformly ;  and  let  A  A'  be  the 
distance  which  the  point  A  is  required  to  traverse  during  a  half  revolution 
of  the  eccentric.  From  the  centre  C,  with  radii  respectively  equal  to  C  A 
and  C  A',  describe  two  circles  ;  divide  the  greatest  into  any  number  of 
equal  parts  (say  16),  and  draw  through  these  points  of  division  the  radii 
C  1,  C  2,  C  3,  &c.  Then  divide  the  line  A  A'  into  the  same  number  of 
equal  parts  as  are  contained  in  the  semicircle  (that  is,  into  8  in  the 
example  now  before  us),  and  through  all  the  points  1',  2',  3',  <fec.,  draw 
circles  concentric  with  the  former ;  the  points  of  their  intersection  B,  D,  E, 
&c.,  with  the  respective  radii  C  1,  C  2,  C  3,  &c.,  are  points  in  the  curve 
required,  its  vertex  being  at  the  point  8. 

It  will  now  be  obvious  that  when  the  axis,  in  its  angular  motion,  shall 
have  passed  through  one  division,  in  other  words,  when  the  radius  C  1 
coincides  with  C  A7,  the  point  A,  being  urged  upwards  by  the  curvature 
of  the  revolving  body  on  which  it  rests,  will  have  taken  the  position  indi- 
cated by  1' ;  and  further,  when  the  succeeding  radius  C  2  shall  have 
assumed  the  same  position,  the  point  A  will  have  been  raised  to  2',  and  so 
on  till  it  arrives  at  A',  after  a  half  revolution  of  the  eccentric.  The  re- 
maining half  A  G  F  8  of  the  eccentric,  being  exactly  symmetrical  with  the 
other,  will  enable  the  point  A  to  descend  in  precisely  the  same  manner  as 
it  is  elevated.  It  is  thus  manifest  that  this  curve  is  fitted  to  impress  a  uni- 


186  DRAWING   OF   MACHINERY. 

form  motion  upon  the  point  A  itself,  but  in  practice  a  small  friction 
roller  is  usually  interposed  between  the  surface  of  the  eccentric  and  the 
piece  which  is  to  be  actuated  by  it.  Accordingly,  the  point  A  is  to  be 
taken  as  the  centre  of  this  roller,  and  the  curve  whose  construction  we 
have  just  explained  is  replaced  by  another  similar  to,  and  equidistant  from 
it,  which  is  drawn  tangentially  to  arcs  of  circles  described  from  the  various 
points  in  the  primary  curve  with  the  radius  of  the  roller.  This  second 
curve  is  manifestly  endowed  with  the  same  properties  as  the  other ;  for, 
supposing  the  point  e,  for  example,  to  coincide  with  A,  if  we  cause  the 
axis  to  revolve  through  a  distance  equal  to  one  of  the  divisions  the  point 
fj  which  is  the  intersection  of  the  curve  with  the  circle  whose  radius  is 
C  1',  will  then  obviously  have  assumed  the  position  V ;  at  the  next  por- 
tion of  the  revolution,  the  point  y  (which  is  such  that  the  angle  f  C  g  is 
equal  to  e  C  f]  will  have  arrived  at  2',  and  so  on.  Thus  it  is  plain  that 
the  point  a  will  be  elevated  and  depressed  uniformly  by  means  of  the 
second  curve,  in  the  same  manner  as  .that  denoted  by  A  is  actuated  by 
tlis  first, 

It  is  obvious  that  the  movable  point  a  must,  in  actual  working,  be  held 
in  contact  with  the  surface  of  the  eccentric  ;  this  is  generally  accomplished 
by  the  action  of  a  weight  or  of  a  spring  ;  but  in  forms  similar  to  fig.  1,  in 
which  all  the  diameters,  as  A  B,  B  F,  D  G,  &c.,  are  equal,  two  frictions 
connected  and  placed  diametrically  opposite  each  other  may  be  used, 
which  will  be  thus  alternately  and  similarly  impelled ;  in  many  cases  an 
eccentric  groove  is  cut,  and  the  friction  roll  or  point  a  is  made  to  slide  in 
this  groove. 

Fig.  2. — To  draw  a  double  eccentric  curve,  which  shall  impart  a  uni- 
form molion  of  ascent  and  descent  to  the  point  A,  traversing  an  arc  of  a 
circle  A  A'. 

First,  divide  the  given  arc  A  A'  into  any  number  of  equal  parts  (8  in 
the  present  example),  and  from  the  common  centre,  or  axis  C  of  the  eccen- 
tric, describe  circles  passing  through  each  of  the  points  of  division  1',  2',  3', 
&c.  Divide  also  the  circle  passing  through  O,  the  centre  of  the  arc  A  A', 
into  twice  the  number  of  equal  parts  ;  then  taking  up  in  the  compasses  the 
length  A  O,  and  placing  one  of  the  points  at  the  division  marked  1,  de- 
scribe an  arc  of  a  circle,  which  will  cut  at  B  the  circle  drawn  with  the 
radius  C  I/;  from  the  next  point  of  division  2,  mark  off,  in  the  same  man- 
ner, the  point  D  in  the  circle  whose  radius  is  C  2',  and  so  on.  The  points 
B,  D,  E,  &c.,  thus  obtained,  are  points  in  the  curve  required,  which,  sup- 
posing the  eccentric  to  revolve  uniformly,  will  possess  the  property  of 
communicating  to  the  point  A  a  uniform  motion  of  ascent  and  de- 


DE AWING    OF   MACHINERY.  187 

scent  along  the  arc  A  A'.  This  admits  of  easy  demonstration.  The  angle 
B  C  1'  is  half  of  2'  C  D,  and  consequently,  when  the  point  B  has  arrived 
at  1',  the  radius  C  D,  then  coinciding  with  C  B,  will  have  passed  through 
an  angle  equal  to  I'  C  B,  and  again,  at  the  next  point  in  the  revolution, 
will  coincide  with  C  2'.  Therefore  the  portion  B  D  of  the  curve  will 
impel  the  given  point  through  the  arc  V  2',  in  the  same  time  and  with  the 
same  velocity,  as  the  part  A  B  will  have  raised  it  from  A  to  1'.  By  a 
similar  process  of  reasoning  it  will  be  manifest,  that  the  angle  1'  C  B  being 
just  one-third  of  3'  C  I,  the  point  A  will  also  traverse  the  space  2'  3'  with 
a  uniform  motion. 

By  a  glance  at  the  figure  it  will  be  seen  that  this  curve  is  not  symmetri- 
cal ;  in  other  words,  that  the  part  A  F  E  is  not  equal  or  similar  to  A  D  E. 
This  may  be  accounted  for  by  observing,  that  the  arc  1)  1',  for  instance,  is 
equal  to  V  B,  and  consequently  the  point  ~b  (which  is  determined  by  the 
intersection  of  the  circle  passing  through  V  with-  the  arc  described  from 
the  centre  15)  cannot  be  situated  in  the  same  position  in  relation  to  A  as 
the  point  B,  since  the  radius  C  A  does  not  pass  through  1' ;  the  same  re- 
mark applies  to  all  the  other  arcs,  d  2',  &c.  It  is  not  the  less  certain, 
however,  that  the  part  A  F  E  of  the  eccentric  will  cause  the  given  point 
to  descend  through  the  arc  A'  A  in  the  same  uniform  manner  as  it  had 
been  elevated  by  the  part  A  D  E. 

In  the  two  preceding  examples  of  eccentrics  it  has  been  shown,  that 
the  point  A  moves  through  equal  spaces  in  equal  times,  both  in  ascending 
and  descending.  In  some  cases,  however,  this  is  by  no  means  desirable  ; 
thus,  if  the  eccentric  is  destined  to  give  motion  to  a  mass  of  matter  which 
offers  considerable  resistance,  such  a  form  would  give  rise  to  injurious  and 
destructive  shocks.  In  such  cases,  it  is  necessary  so  to  regulate  the  curva- 
ture of  the  eccentric,  that  the  point  A  shall  move  at  the  beginning  and 
end  of  its  stroke  with  diminished  velocity ;  and  that  for  this  purpose, 
the  space  A  A  should  be  unequally  divided,  as  in  the  example  which 
comes  next  under  notice. 

Fig.  3. — To  draw  a  double  and  symmetrical  eccentric  curve,  suck  as  to 
cause  the  point  A  to  move  in  a  straight  line,  and  with  an  unequal  motion  ; 
the  velocity  of  ascent  being  accelerated  in  a  given  ratio  from  the  starting 
point  to  the  vertex  of  the  curve,  and  the  velocity  of  descent  being  retarded 
in  the  same  ratio. 

Upon  A  A'  as  a  diameter  describe  a  semicircle,  and  divide  it  into  any 
number  of  equal  parts  ;  draw  from  each  point  of  division  1',  2',  3',  &c., 
perpendiculars  upon  C  A';  and  through  the  points  of  intersection  I2,  22,  3s, 
&c.,  draw  circles  having  for  their  common  centre  the  point  C,  which  is  to 


188  DRAWING   OF   MACHINERY. 

be  joined,  as  before,  to  all  the  points  of  division  on  the  circle  (A'  48.) 
The  points  of  intersection  of  the  concentric  circles  with  the  radii  C  1,  C  2, 
C  3,  &c.,  are  points  in  the  curve  required. 

Fig.  4. — To  construct  a  double  and  symmetrical  eccentric,  which  shall 
produce  a  uniform  rectilinear  motion,  with  periods  of  rest  at  the  points 
nearest  to,  and farthest  from,  the  axis  of  rotation. 

The  lines  in  the  figure  above  referred  to  indicate  sufficiently  plainly, 
without  the  aid  of  further  description,  the  construction  of  the  curve  in 
question,  which  is  simply  a  modification  of  the  eccentric  represented  at 
Fig.  1.  In  the  present  example,  the  eccentric  is  adapted  to  allow  the 
movable  point  A  to  remain  in  a  state  of  rest  during  the  first  quarter  of  a 
revolution  B  D  ;  then,  during  the  second  quarter,  to  cause  it  to  traverse, 
with  a  uniform  motion,  a  given  straight  line  A  A',  by  means  of  the  curve 
D  G ;  again,  during  the  next  quarter  E  F  G,  to  render  it  stationary  at  the 
elevation  of  the  point  A! ;  and  finally,  to  allow  it  to  subside  along  the 
curve  B  E,  with  the  same  uniform  motion  as  it  was  elevated,  to  its  original 
position,  after  having  performed  the  entire  revolution. 

Fig.  5  represents  an  edge  view  of  this  eccentric,  and  fig.  6  a  vertical 
section  of  it. 

Figs.  7,  8,  and  9,  a  Circular  Eccentric. — These  figures  represent  a 
model  of  a  variety  of  the  circular  eccentric,  which  is  the  contrivance 
usually  adopted  in  steam-engines  for  giving  motion  to  the  valves  regu- 
lating the  action  of  the  steam  upon  the  piston.  The  circular  eccentric 
is  simply  a  species  of  disc  or  pulley  fixed  upon  the  crank-shaft,  or  other 
rotating  axis  of  an  engine,  in  such  a  manner  that  the  centre  or  axis 
of  the  shaft  shall  be  at  a  given  distance  from  the  centre  of  the  pulley.  A 
ring  or  hoop,  either  formed  entirely  of,  or  lined  with  brass  or  gun  metal, 
for  the  purpose  of  diminishing  friction,  is  accurately  fitted  within  project- 
ing ledges  on  the  outer  circumference  of  the  eccentric,  so  that  the  latter 
may  revolve  freely  within  it ;  this  ring  is  connected  by  an  inflexible  rod 
with  a  system  of  levers,  by  which  the  valve  is  moved.  It  is  evident,  that 
as  the  shaft  to  which  the  eccentric  is  fixed  revolves,  an  alternating  recti- 
linear motion  will  be  impressed  upon  the  rod,  its  amount  being  determined 
by  the  eccentricity,  or  distance  between  the  centre  of  the  shaft  and  that 
of  the  exterior  circle.  The  throw  of  the  eccentric  is  twice'the  eccentricity 
C  E  ;  or  it  may  be  expressed  as  the  diameter  of  the  circle  described  by 
the  point  E.  The  nature  of  the  alternating  motion  generated  by  the  cir- 
cular eccentric  is  identical  with  that  of  the  crank,  which  might  in  many 
cases  be  advantageously  substituted  for  it. 

Fig.  8  is  the  edge  view,  fig.  9  the  section  of  the  eccentric,  in  this  par- 


DRAWING    OF   MACHINERY.  1$9 

ticular  example,  formed  in  a  single  piece,  and  which  can  be  applied  only 
when  the  shaft  to  which  it  is  to  be  attached  is  straight  and  uninterrupted 
by  cranks,  <fec.  The  mode  of  representing  the  arm  in  fig.  9,  which  is  a 
section  on  the  line  D  F,  is  not  strictly  accurate,  but  is  a  license  frequently 
practised  in  similar  cases,  and  which  is  attended  with  obvious  advantage. 

In  many  machines,  the  eccentric  is  used  for  the  raising  of  a  weight  a 
certain  height  and  then  letting  it  fall,  as  in  the  case  of  ore  stampers,  cloth 
beetles,  trip  hammers,  and  the  valve  rod  of  some  steam  engines.  In  these 
cases  the  eccentric  may  be  considered  as  merely  a  single  long  tooth  geer, 
in  which  commonly,  on  account  of  the  uniformity  of  action,  the  wiping  or 
rubbing  surface  is  an  involute  curve,  the  boss  of  the  eccentric  being  the 
generating  circle. 

In  practice,  the  term  eccentric  is  generally  confined  to  the  circular 
eccentric  ;  all  others,  with  exception  of  that  last  described,  or  wypers, 
being  called  cams. 

DRAWING     OF     SCREWS. 

The  screw  is  a  cylindrical  piece  of  wood  or  metal,  in  the  surface  of 
which  one  or  more  helical  grooves  are  formed.  The  thread  of  the  screw 
is  the  solid  portion  left  between  the  grooves  ;  and  the  pitch  of  the  screw. 
is  the  distance,  measured  on  a  line  parallel  to  the  axis  of  the  cylinder,  be- 
tween the  two  contiguous  centres  of  the  same  thread. 

Projections  of  a,  triangular-threaded  screw  and  nut,  pi.  XXVIII.,  fig.  1. 
— Having  drawn  the  ground  line  A  B,  and  the  centre  lines  C  C'  of  the 
figures,  from  O  as  a  centre,  with  a  radius  equal  to  that  of  the  exterior 
cylinders,  describe  the  semicircle  a  3  6  ;  describe  in  like  manner  the  semi- 
circle 1)  c  e  with  the  radius  of  the  interior  cylinder.  Now  draw  the  per- 
pendiculars a  a"  and  6  6",  T)  V  and  e  c",  which  will  represent  the  ver- 
tical projections  of  the  exterior  and  interior  cylinders.  Then  divide  the 
semicircle  a  3  6  first  described  into  any  number  of  equal  parts,  say  6,  and 
through  each  point  draw  radii,  which  will  divide  the  interior  semicircle 
similarly.  On  the  line  a'  a"  set  off  the  length  of  the  pitch  as  many  times 
as  may  be  required ;  and  through  the  points  of  division  draw  straight 
lines  parallel  to  the  ground  line  A  B.  Then  divide  each  distance  or  pitch 
into  twice  the  number  of  equal  parts  that  the  semicircles  have  been  divided 
into,  and  following  instructions  already  laid  down  (page  100),  construct 
the  helix  a'  3'  6  both  in  the  screw  and  nut. 

Having  obtained  the  point  V  by  the  intersection  of  the  horizontal  line 
passing  through  the  middle  division  of  a!  a  with  the  perpendicular  b  V ,  de- 


190  DRAWING   OF   MACHINERY. 

scribe  the  helix  V  c'  e ',  which  will  represent  the  bottom  of  the  groove. 
The  apparent  outlines  of  the  screw  and  its  nut  will  then  be  completed  by 
drawing  the  lines  V  a!,  a'  I',  etc.,  to  the  curves  of  the  helices ;  these 
are  not,  strictly  speaking,  straight  lines,  but  their  deviation  from  the 
straight  line  is,  in  most  instances,  so  small  as  to  be  imperceptible,  and  it 
is  therefore  unnecessary  to  complicate  the  drawing  by  introducing  the 
method  of  determining  them  with  rigorous  exactness. 

When  a  long  series  of  threads  have  to  be  delineated,  they  should  be 
drawn  mechanically  by  means  of  a  mould  or  templet,  constructed  in  the 
following  manner : — Take  a  small  slip  of  thin  wood  or  pasteboard,  and 
draw  upon  it  the  helix  a'  3'  G  to  the  same  scale  as  the  drawing,  and  pare 
the  slip  carefully  and  accurately  to  this  line.  By  applying  this  templet 
upon  fig.  1,  so  that  the  points  a!  and  6  on  the  plate  shall  coincide  with 
#'and-6  on  the  drawing,  the  curve  a'  3'  6  can  be  drawn  mechanically, 
and  so  on  for  the  remaining  curves  of  the  outer  helix.  The  same  templet 
may  be  employed  to  draw  the  corresponding  curves  in  the  screw-nut  by 
simply  inverting  it ;  but  for  the  interior  helix  a  separate  one  must  be  cut, 
its  outlines  being  laid  off  in  the  same  manner. 

Projections  of  a  square-threaded  screw  and  nut  (fig.  2). — The  depth  of 
the  thread  is  equal  to  its  thickness,  and  this  latter  to  the  depth  of  the 
groove.  The  construction  is  similar  to  the  preceding,  and  will  be  readily 
understood  from  the  drawing,  the  same  letters  and  figures  marking  rela- 
tive parts.  The  parts  of  the  curve  concealed  from  view  are  shown  in 
dotted  lines. 

It  will  be  observed,  that  the  heads  and  nuts  of  the  screws  are  repre- 
sented as  broken,  which  is  done  for  economy  of  space. 

It  is  s.eldom  necessary  to  delineate  so  exactly  the  outlines  of  screws, 
as  they  are  generally  drawn  to  a  much  smaller  scale.  Fig.  3  shows  the 
simplest  form  by  which  a  screw  may  be  represented.  Figs.  4,  5,  6,  repre- 
sent a  triangular-threaded,  a  square-threaded  screw,  and  a  serpentine,  in 
which  the  helical  curves  are  replaced  by  straight  lines,  and  these  forms 
will  be  found  sufficiently  exact  and  graphic  for  most  of  the  cases  occur- 
ring in  practice. 

Screws  may  have  two,  three,  or  even  a  greater  number  of  threads,  ac- 
cording to  the  velocity  which  their  action  may  be  required  to  produce. 
A  double-threaded  screw  is  one  in  which  the  pitch  of  any  individual  helix 
includes  two  threads ;  a  three-threaded  screw,  one  in  which  it  embraces 
three  threads,  and  so  on. 

Size  and  proportion  of  bolts. — The  diameter  of  the  bolt  depends,  of 
course,  on  the  strain  to  which  it  is  to  be  subjected  ;  but  since  the  tensile 


DRAWING    OF   MACHINERY. 


191 


strength  of  common  bolts  is  reduced  at  least  one  quarter  by  the  cutting  of 
the  thread,  as  a  safe  rule  the  tension  ought  not  to  exceed  4  tons  on  the 
square  inch  of  section.  It  will  be  found  economical  often,  where  the  bolt 
is  long,  to  cut  the  thread  on  a  larger  wire,  and  weld  the  piece  to  a  rod  of 
the  interior  diameter  of  the  screw.  The  section  of  the  thread  most  ap- 
proved of  for  strength  and  easy  motion  of  the  nut  is  the  equilateral  tri- 
angle, thus  A ,  the  bevelled  .sides  being  equal  between  themselves  and  to 
the  base. 


Diameter  of 
bolts. 

Threads  in 
inch. 

Short  diameter 
of  nut. 

Diameter  of 
bolts. 

Threads  in 
inch. 

Short  diameter 
of  nut 

Inches. 

Inches. 

Inches. 

Inches. 

1 

12 

I 

If 

5 

»l 

5 

11 

.  irV 

11 

** 

8| 

i 

10 

!A 

2 

*l 

»iV 

7 
? 

8? 

IT"* 

2J 

4 

8| 

1 

8 

If 

21 

4 

4 

M 

n 

2 

II 

8* 

4} 

1* 

7 

21 

2* 

8} 

*A 

»l 

6 

V* 

2| 

3 

4? 

H 

51 

»tt 

2? 

3 

*H 

If 

5 

sj 

3 

3 

*i 

The  thickness  of  the  nut  should  be  equal  to  the  diameter  of  the  bolt. 
The  head  of  the  bolt  is  usually  square  ;  the  nut  may  be  of  the  same  form, 
but  as  often  is  six-paned  or  six  square.  When  the  head  of  the  bolt  is  in- 


Fig.  280. 


Fig.  281. 


tended  to  \>Q  flush  or  even  with  the  surface  of  the  piece  into  which  the  bolt 
is  inserted,  the  inside  of  the  head  is  made  conical  like  the  common  wood- 


192  DRAWING   OF   MACHINERY. 

screw,  or  pyramidal,  and  it  is  then  said  to  be  countersunk.  "When  bolts 
are  employed  in  wood,  washers  are  usually  placed  beneath  the  nut  and 
head,  to  give  a  more  extended  bearing  surface. 


Figs.  280  and  281  represent  two  wrought  iron  hooks,  in  which  the 
material  is  distributed  according  to  the  strain  to  which  the  parts  may  be 
subjected.  The  following  are  the  proportions  on  which  fig.  280  is  con- 
structed : — Assuming  the  neck  of  the  hook  as  the  modulus  or  1,  the  diam- 
eter of  journals  of  the  traverse  are  1.1  ;  width  of  traverse  at  centre,  2  ; 
distance  from  the  centre  of  the  hook  to  the  centre  of  the  traverse,  7.5 ; 
interior  circle  of  the  hook,  3.4 ;  greatest  thickness  of  the  hook,  2.8.  As- 
suming (fig.  281)  the  diameter  of  the  wire  of  the  chain  as  1 :  interior  circle 
of  hook  is  3.2,  and  greatest  thickness  of  hook,  3.5. 

/ 

FRAMES. 

Plate  XXX.  represents  the  application  of  iron  in  the  frames  of  tools. 

Fig.  1  represents  the  cam-punch  and  shear;  in  this  case,  the  force 
exerted  whilst  the  machine  is  in  the  operation  of  punching  or  shearing, 
tends  to  open  the  jaws  a  aj  and  the  tendency  increases  with  the  depth  of 
the  jaw,  the  strain  obviously  being  the  greatest  at  the  inmost  part  of  the 
jaw.  The  frame  consists  of  a  plate  of  cast  iron,  with  two  webs  around  its 
edges;  the  front  web  being  subjected  to  a  tensile  strain,  should  be  in  the 
area  of  its  section  about  six  times  that  of  the  rear  web  which  is  subjected 
to  a  cornpressive  force. 

Fig.  2  is  the  side  frame  of  a  planing  machine.  The  force  here  exerted 
is  horizontal  against  the  cutter,  which  can  be  raised  or  lowered  at  pleasure, 
according  to  the  magnitude  of  the  work  to  be  planed ;  the  upright  has, 
therefore,  to  be  braced,  which  is  done  in  a  curved  form  for  beauty  of  out- 
line. 

Fig.  3  is  a  common  jack-screw,  in  which  the  pressure  is  vertical ;  the 
base  is  made  extended  to  give  it  stability. 

Fig.  4:  is  a  plan  of  the  top  plate,  and  fig.  5  the  elevation  of  a  hydraulic 
press.  The  top  and  bottom  plates  and  platen  are  cast  iron,  the  four  rods 
are  wrought  iron  ;  the  strain  upon  the  rods  is  tensile,  and  it  is  only  neces- 
sary to  give  them  such  a  size  as  to  resist  securely  the  power  which  maybe 
required  on  the  press.  The  plates  are  beams,  supported  at  tiie  four  cor- 
ners ;  subjected  to  a  breaking  strain,  it  will  be  evident  that  the  bottom 


DRAWING   OF   MACHINERY.  193 

plate  is  the  strongest,  as  in  this  case  the  bottom  of  the  plate  being  sub- 
jected to  a  tensile  strain,  is  a  flange  or  platen,  and  affords  more  material 
remote  from  the  neutral  axis  than  the  ribs  of  the  upper  plate  which  are 
subjected  to  the  tensile  strain.  The  movable  platen  is  braced  by  triangu- 
lar wings  or  flanges  radial  from  the  piston.  In  this  particular  case  the 
cylinder  is  cast  iron  hooped  with  wrought  iron ;  it  is  very  common  to 
make  the  whole  cylinder  wrought  iron. 

Principle  of  the  action  of  the  hydrostatic  press. — Let  A  B  C  D  (fig.  282) 
epresent  a  vertical  section  of  a  cylindrical  vessel  filled  with 
an  incompressible  non-elastic  fluid,  as  water  for  instance  ;  let 
E  and  F  be  two  pistons  of  different  magnitudes  connected  with 
the  cylinder,  and  fitting  closely  their  respective  orifices ;  now, 
whatever  pressure  be  exerted  by  the  piston  F  on  the  fluid  Fis- 2S2< 
in  the  cylinder,  it  will  be  repeated  on  the  piston  E  as  many  times  as  the 
area  of  the  small  piston  is  contained  in  the  large  piston ;  that  is,  if  the 
area  of  F  was  1  square  inch,  and  the  pressure  exerted  10  Ibs.,  and  the  pis- 
ton E  100  square  inches,  then  the  pressure  on  E  would  be  10  x  100,  or 
1000  Ibs.  F  corresponds  to  the  plunger  of  the  force-pump,  E  to  the  piston 
or  ram  of  the  press.  The  thickness  of  metal  of  the  cylinder,  if  of  cast-iron, 
should  not  be  less  than  one-half  the  diameter  of  the  ram.  Adopting  this 
as  the  rule,  to  find  the  entire  pressure  in  tons  which  -a  cylinder  can  sustain, 
the  diameter  of  the  ram  being  given : 

Multiply  the  square  of  the  diameter  in  inches  by  3,  and  the  product 
will  be  the  pressure  in  tons. 

Or,  the  pressure  in  tons  being  given  : 

Divide  the  given  pressure  in  tons  by  3,  and  the  square  root  of  the  quo- 
tient will  be  the  diameter  of  the  piston  in  inches.  Tims,  the  diameter  of 
the  piston  being  10  inches,  the  thickness  of  metal  5  inches,  the  pressure 
might  be  10  x  10  x  3  =  300  tons. , 

Figs.  6  &  7  represent  a  housing  for  rolls.  The  screw  a  presses  down 
upon  the  top  of  the  box  of  the  journal,  and  the  effect  is  a  tensile  strain  on 
the  sides  of  the  frame ;  but  it  must  be  remarked,  that  frames  of  this  sort 
are  subject  to  percussive  and  intermittent  strain  vastly  exceeding  the  mere 
tensile  strain,  and  proper  allowance  is  to  be  made  for  this ;  and  it  is  much 
better  to  depend  in  part  on  mass  or  on  dead  weight  of  material  to  resist 
such  strains  than  upon  cohesive  strength  merely. 

PL  XXXI.  represents  the  elevation  of  the  frames  of  three  classes  of 
American  marine  engines. 

Figs.  1  and  2  represent  the  frame-work  of  the  New  World.     It  is  com- 
posed of  four  pieces  of  heavy  pine  timber  dd,  which  are  formed  into  two* 
13 


194  DRAWING   OF   MACHINERY. 

triangles,  and  inclined  slightly  laterally  to  each  Other  (fig.  2) ;  their  lower 
ends  rest  on.  the  keelsons,  and  upon  their  upper  extremities  are  placed  the 
pillow-block  c  of  the  working  beam.  They  are  solidly  fastened  together 
and  to  the  boat  by  numerous  horizontal  and  diagonal  timbers,  which  are 
secured  by  wooden  knees  and  keys,  and  are  heavily  bolted.  The  two  front 
legs  are  bolted  to  flanges  cast  on  the  sides  of  the  condenser,  and  the  other 
end  of  the  framing  is  attached  to  a  large  mass  of  timbers,  which  support 
the  shaft  pillow-block  &.  The  framing  is  further  steadied  by  two  addi- 
tional timbers,  and  rods  running  from  the  beam  pillow-blocks  outside  the 
shaft  to  the  keelsons  of  the  boat ;  a  represents  the  guides,  which  are  bolted 
at  the  bottom  to  the  cylinder  flange,  and  retained  in  their  vertical  position 
by  wrought  iron  braces  connected  with  the  framing.  The  entire  fastening 
of  the  engine  and  its  framing  is  so  disposed  as  to  reduce  all  the  strains  to 
direct  ones  of  extension  or  compression  on  the  fibres  of  the  iron  and  wood 
employed  in  the  construction.  The  height  of  the  frame  is  46  feet,  width 
at  bottom  31  feet. 

Fig.  3  represents  the  side  elevation  of  the  frame  of  the  side  lever  ocean 
steamer  Pacific.  In  this  frame  the  two  large  hollow  pillow-blocks  which 
sustain  the  shaft  on  each  side  of  the  cranks  are  supported  by  four  wrought 
iron  columns  G  G  on  the  forward  extremity  of  the  bed-plate  I,  the  centre 
of  the  shaft  being  23  feet^above  the  keelson.  The  pillow-blocks  thus  sup- 
ported are  connected  by  two  strong  inclined  braces  D  to  the  cylinder,  by 
means  of  solid  facings  cast  with  it  on  each  side  of  the  steam  opening.  The 
columns  are  connected  by  horizontal  braces  A  A,  composed  of  hollow  tubes, 
through  which  bolts  pass,  and  the  frames  of  the  two  engines  are  connected 
at  the  same  points  by  similar  tubes,  and  also  two  diagonal  horizontal  braces 
cast  together.  Similar  braces  C  C  are  used  to  connect  each  extremity  of 
the  pillow-blocks,  and  the  two  engine  frames  are  connected  by  a  horizontal 
wrought  iron  cross.  To  resist  the  tendency  of  the  engines,  in  the  rolling 
of  the  ship,  to  press  the  outer  bearings,  there  are  in  a  vertical  transverse 
plan  three  wrought  iron  cross  or  diagonal  braces  F  between  the  pillow- 
blocks  and  bed-plates.  Four  cross  braces  II  and  J  connect  the  extremities 
of  the  cylinder  and  the  frame.  The  cylinders  are  also  connected  by  a 
horizontal  tubular  brace.  It  will  be  thus  seen  that  this  frame  is  a  sys- 
tem of  bracing  and  cross-bracing,  in  which  the  material  is  most  economi- 
cally disposed  to  resist  the  various  strains. 

The  bed-plate  consists  of  a  single  casting,  32  feet  long  and  9  feet  broad, 
which  is  securely  fastened  to  the  keelsons  and  ship's  bottom  ;  the  diameter 
of  the  cylinder  is  9p  inches,  and  the  stroke  9  feet. 

Fig.  4  represents  the  side  view  of  the  frame  of  the  inclined  engines  of 


DRAWING    OF   MACHINERY.  195 

the  war  steamer  Susquehanna.  The  cylinder  A  is  rested  between  two  tri- 
angular frames,  on  the  inclination  of  the  longest  side,  and  is  securely  bolted 
to  each  frame.  The  two  frames  are  connected  together  with  braces  similar 
to  those  of  the  Pacific,  and  the  whole  is  securely  bolted  on  to  the  keelsons 
and  bottom  of  the  ship.  In  this  illustration,  the  main  pieces  of  the  frame 
are  made  of  boiler  iron,  constructed  like  box  girders  ;  but  in  smaller  en- 
gines, it  is  usual  to  make  these  parts  of  wood.  The  diameters  of  the  cylin- 
ders are  5  feet  10  inches,  the  length  of  stroke  10  feet. 

PL  XXXII. — Fig.  1  represents  the  working-beam  of  the  New  World. 
It  is  composed  of  a  skeleton  frame  of  cast  iron,  round  which  a  wrought 
iron  strap  A  is  fixed.  This  strap  is  forged  in  one  piece,  and  its  extreme 
ends  are  formed  into  large  eyes,  which  are  bored  out  to  receive  the  end 
journals.  The  skeleton  frame  is  a  single  casting,  and  contains  the  eyes  for 
the  main  centre  and  air-pump  journal ;  the  centre  hub  is  strengthened  by 
wrought  iron  hoops  a  a,  which  are  shrunk  upon  it.  At  the  points  of  con- 
tact of  the  strap  and  skeleton,  key-beds  are  prepared,  into  which  the  keys 
are  carefully  fitted  and  tightly  driven ;  the  keys  are  afterwards  riveted 
over  at  both  ends,  which  retains  them  in  their  places,  as  well  as  the  strap 
on  the  skeleton  frame.  The  strap  is  also  secured  to  the  frame  by  straps 
5  5  and  keys.  The  skeleton  frame  is  still  further  braced  by  wrought  iron 
straps  C  C,  which  tie  the  middle  of  the  long  arms  of  the  cross  to  the  ex- 
tremities of  the  shorter  ones.  This  form  of  beam  is  that  usually  adopted 
for  the  engines  of  eastern  American  river  boats ;  the  proportions  are  some- 
what varied,  but  the  form  is  identical.  The  following  are  the  dimensions 
of  our  illustration: — From  centre  to  centre  of  end  journals,  26  feet;  this 
is  somewhat  less  than  the  usual  proportion  to  length  of  stroke,  being  but 
slightly  less  than  double  the  stroke ;  length  of  centre  hub,  26  inches  ; 
diameter  of  main  centre  eye,  15 f  ;  of  eye  for  air-pump  journal,  6| ;  of  end 
journal,  8}  inches. 

Fig.  3  represents  a  side  elevation ;  fig.  4,  a  plan  ;  and  fig.  5  a  section  of 
a  cast  iron  working  beam  of  an  English  stationary  engine.  It  will  be  per- 
ceived that  the  outline  of  the  beam  is  a  parabola,  it  being  in  effect  a  beam 
supported  at  the  centre  and  loaded  at  the  extremities. 

From  the  following  table  of  practical  examples  from  "  Architecture  of 
Machinery,"  we  would  assume  as  a  safe  rule  for  land  engines,  that  the 
depth  at  centre  should  be  the  diameter  of  the  cylinder,  and  the  length  of 
beam  three  times  the  length  of  stroke.  Hence  we  can  construct  the  out- 
line, having  for  the  vertex  the  extremity  of  the  beam  and  the  point  B  in 
the  curve  at  the  centre.  The  sectional  area  may  be  estimated  from  rules 
already  given,  knowing  the  load  at  the  extremity,  that  is ;  the  pressure  on 


196 


DRAWING   OF   MACHINERY. 


the  piston,  the  weight  of  the  same  and  its  connections,  and  also  the  force 
required  to  drive  the"  air-pump,  estimated  at  the  extremity  of  the  lever. 
As  an  engine  is  subject  to  shocks,  the  load  should  be  estimated  as  six 
times  the  absolute  load.  "  Five  per  cent,  of  the  nominal  power  of  the 
engine  may  be  considered  the  maximum  of  power  required  to  drive  the 
air-pump." — Ed.  Tredgold. 


Diameter  of 
cylinder. 

Length  of  stroke. 

Description  of 
work. 

Length  of  beam 
from  centre. 

Depth  at  centre. 

Sectional  area. 

inches. 

ft.  in. 

ft.   in. 

inches. 

square  inches. 

4TJ 

s 

Rolling, 

12    4 

43 

240 

401 

7 

Pumping, 

10    4 

36 

162 

39* 

6    9 

Blowing 

9     6 

85* 

96i 

864 

6    3 

Boiling, 

9    3 

80 

60 

84 

5 

Mill  work, 

8 

25 

50 

W 

4 

" 

6  10 

22J 

50 

42 

4 

Marine, 

6    3 

23 

133 

42 

4    2 

« 

G    6 

2T 

216 

32 

3 

11 

5 

22 

132 

1 

Figs.  6  and  7  represent  a  side  and  a  front  elevation  of  a  crank,  such 
as  is  usually  adopted  on  the  engines  of  American  river  boats.  The  main 
body  of  the  crank  is  of  cast  iron,  with  two  horns 
a  a  projecting  from  the  central  hub,  and  the 
whole  is  bound  with  a  strap  of  wrought  iron. 
It  is  evident  that  this  form  of  crank  gives  the 
greatest  amount  of  strength  with  the  least  ma- 
terial, and  belongs  to  the  same  class  of  construc- 
tion as  the  working  beam  (fig.  1).  The  eye  of 
the  crank  is  usually  made  one-fourth  the  diam- 
eter of  the  cylinder.  The  table  from  Eedten- 
bacher  here  inserted  gives  the  relative  sizes  of 
central  and  end  eyes  of  cranks,  depending  on 
the  proportion  between  the  length  of  crank  and 
the  diameter  of  central  eye.  The  first  column 


ni.VMETEK   OP   EYE,    BEING   UNITS. 

For  wrought 
iron  shaft. 

Cast  iron 
shafts. 

2 

0.85 

0.62 

3 

0.69 

0.51 

4 

0.60 

0.44 

5 

0.54 

0.39 

6 

0.49 

0.36 

1 

-.       0.45 

0.33 

8 

0.42 

0.31 

9 

0.40 

0.23 

10 

0.3S 

0.23 

11 

0.36 

0.26 

12 

0.34 

0.25 

13 

0.33 

0.24 

exhibits  the  number  of  times  the  diameter  of  eye  is  contained  in  the  length 
of  crank ;  the  other  columns  exhibit  the  diameter  of  crank-pin. 

From  this  table  may  be  determined  for  any  crank  the  diameter  of 
either  eye,  one  being  known,  and  the  length  of  the  crank. 

Figs.  8,  9,  a  side  view  and  front  elevation  of  a  wrought  iron  crank  and 
their  practical  proportions  ;  the  eye  for  the  crank-pin  is  a  slightly  conical 
hole,  and  the  pin  is  made  of  a  corresponding  taper. 

PI.  XXXIII.  represents  steam-engine  connecting-rods  and  their  details. 

Figs.  1,  2,  represent  the  front  and  side  elevation  of  a  cast  iron  connect- 


DRAWING    OF   MACHINERY.  197 

ing-rod.  It  is  a  bar,  strengthened  throughout  the  greater  part  of  its  length 
by  four  ribs  or  feathers,  whose  outlines  .in  the  direction  of  their  length  are 
parabolic  curves.  Its  upper  extremity  is  formed  into  two  projecting  arms, 
upon  each  of  which  a  close  wrought  iron  strap  A  is  fixed  by  means  of  a 
key  or  cotter  c.  These  straps  are  provided  and  formed  for  the  reception 
of  the  brass  bushes  a  and  J,  which  are  accurately  fitted  to  the  journals  or 
bearings  of  the  cross  head. 

The  lower  end  of  the  connecting-rod  is  made  of  a  form  suitable  for  the 
reception  of  the  brasses,  and  other  adjusting  mechanism  necessary  for  the 
purpose  of  acting  freely,  but  without  play,  upon  the  pin  of  the  crank. 
In  the  present  example,  the  end  of  the  crank-pin  is  concealed  by  a  slight 
brass  cover  or  disc,  fixed  to  the  connecting-rod  by  two  small,  screw  pins, 
which  serves  to  protect  the  working  surfaces  from  dust,  and  imparts  an 
elegant  finish  to  the  whole. 

In  fig.  3  the  ends  of  the  connecting-rod  are  represented  upon  a  scale 
of  double  the  magnitude  of  the  preceding  figures.  One  of  the  upper  links 
or  straps,  with  its  adjusting  apparatus,  is  supposed  to  be  cut  by  a  vertical 
plane  passing  through  the  axes,  so  as  to  expose  the  interior  arrangement. 
This  section  exhibits  distinctly  the  mode  of  fixing  the  links  upon  the  arms 
of  the  connecting-rod  by  means  of  the  cotters  <?,  c,  and  projecting  discs  e,  e, 
cast  upon  the  arms  ;  as  also  the  contrivance  for  retaining  the  brasses  a  and 
1)  in  their  places.  Fig.  5,  which  is  a  vertical  section,  shows  the  corre- 
sponding provisions  for  the  lower  end  of  the  rod  ;  a  small  oblique  hole  for 
the  introduction  of  oil  will  be  observed  in  the  upper  brass  my  while  the 
lower  n  is  formed  with  a  spherical  projection  entering  a  concave  recess  in 
the  cast  iron,  for  the  purpose  of  preventing  its  displacement  by  the  friction 
of  the  crank-pin,  which  is  regulated  and  adjusted  by  the  cotter  d. 

Fig.  6  is  a  horizontal  section  on  the  line  a  J,  showing  the  form  of  the 
body  and  feathers  of  the  rod. 

Fig.  4  represents  the  end  of  a  connecting-rod,  in  which  the  arrange- 
ment for  tightening  the  brasses  consists  of  a  gib  b  and  cotter  a.  The  small 
end  of  the  cotter  is  made  with  a  screw,  which  passing  through  a  lug  on 
the  gib,  is  fitted  with  a  nut,  by  means  of  which  the  cotter  is  adjusted  and 
retained  in  any  position  required. 

Figs.  Y  and  8  represent  the  side  and  front  elevation  of  a  wrought  iron 
connecting-rod,  such  as  are  generally  used  on  American  river  boat  engines. 
The  extremities  are  fitted  with  brasses,  straps,  gibs,  and  cotters,  similar 
to  those  already  described.  The  peculiarity  over  the  general  English  con- 
struction is  the  economy  of  material,  and  the  means  adopted  to  give  the 
required  stiffness.  It  consists  of  a  double  truss  brace  a  a  of  round  iron, 


198  DRAWING   OF   MACHINERY. 

which  is  fastened  by  bolts  to  the  rod  near  each  end  ;  struts  5  J,  cut  with  a 
screw,  and  furnished  with  nuts  pass  through  the  centre  of  the  brace,  by 
which  means  the  braces  are  tightened. 

The  length  of  connecting-rods,  as  recommended  "by  English  mechanics, 
is  three  times  that  of  the  stroke  ;  in  this  country  shorter  connecting-rods 
are  used,  twice  the  length  of  stroke  being  not  an  unusual  proportion.  The 
connecting-rod  at  its  smallest  part  near  the  extremities  is  of  the  same  diam- 
eter as  the  piston-rod ;  the  boss  in  the  centre  is  from  1  to  2  inches  more. 

ON     THE     LOCATION     OF     MACHINES. 

In  the  arrangement  of  a  manufactory  or  workshop,  it  is  of  the  utmost 
importance  to  know  how  to  place  the  machinery,  both  as  to  economy  of 
space  and  also  of  working.  "Where  a  new  building  is  to  be  constructed  for 
a  specific  purpose  of  manufacture,  it  will  be  found  the  best  to  arrange  the 
necessaiy  machines  as  they  should  be,  and  then  build  the  edifice  to  suit 
them.  For  defining  the  position  of  a  machine,  we  merely  need  in  outline 
the  space  it  occupies  in  plan  and  elevation,  and  the  position  of  the  driven 
pulley  or  geer,  and  of  the  operative.  To  illustrate  this  subject,  we  have 
selected  a  two- story  weaving  room,  of  which  fig.  283  is  an  elevation  and 
plates  XXXIY,  and  XXXV.  plans. 

In  this  example  the  building  is  rectangular,  of  a  width  and  length  to 
accommodate  the  required  machinery.  The  illustration  is  confined  to  a 
few  rooms  in  one  angle,  the  rest  being  but  a  repeat  of  the  same.  The  tim- 
bering and  planking  are  the  same  as  adopted  at  all  our  large  manufacturing 
places.  Beams  14  to  16  inches  deep,  and  of  little  less  width,  placed  from 
8  to  9  feet  apart  from  centre  to  centre,  and  floored  with  3  to  4  inch 
plank  dowelled  or  matched,  with  top  floors  and  bottom  sheathing.  The 
form  of  construction  being  fixed,  and  the  size  of  the  building  being  deter- 
mined for  the  number  of  looms,  knowing  the  space  they  require  for  the 
machines  and  the  alley  ways ;  lay  down  the  outlines  of  the  building,  and 
dot  in,  or  draw  in  red  or  blue,  the  position  and  width  of  beams.  This  last 
is  of  importance,  as  it  will  be  observed  (fig.  283),  that  no  driving-pulley 
can  come  beneath  the  beam,  and  also  that  this  is  the  position  for  the 
hanger.  Lay  off  now  the  width  of  the  alleys  and  of  the  machines.  The 
first  alley,  or  nearest  the  wall,  is  a  back  alley ;  that  is,  where  the  operative 
does  not  stand,  and  so  on  alternate  alleys.  Draw  the  lines  of  shafting  cen- 
tral to  the  alleys,  as  in  this  position  the  belts  are  least  in  the  way.  One 
operative  usually  tends  four  looms  ;  they  are  therefore  generally  arranged' 
in  sets  of  four,  two  on  each  side  of  the  alley,  being  placed  as  close  to  each 


DRAWING    OF   MACHINERY. 


199 


other  as  possible,  say  one  inch  between  the  lathes,  a  small  cross  alley  being 
left  between  them  and  the  next  set.  Lay  off  now  the  required  alley  at  the 
end  of  the  room,  and  space  oif  the  length  of  two  rows  of  looms  with  alleys 
at  the  end  of  alternate  looms,  and  mark  the  position  of  the  pulleys.  It 


1   i   1   1   J   1   1   j  j    J   1   I 


will  be  observed  that  looms  are  generally  rights  and  lefts,  so  that  the 
pulleys  of  both  looms  come  in  the  space  where  there  is  no  alley.  Should 
the  pulley  come  beneath  a  beam,  the  loom  must  be  either  moved  to  avoid 


2.00  DRAWING   OF   MACHINERY. 

it,  or  the  pulley  may  be  shifted  to  the  opposite  end  of  the  loom.  Parallel 
with  the  pulleys  on  the  looms  draw  the  driving-pulleys  on  the  shafts,  that 
is,  ~k  parallel  with  &,  I  with  i,  /  with  /,  and  so  on.  Proceed  now  to  draw 
the  third  and  fourth  row  of  looms,  since  the  second  and  third  rows  are 
driven  from  the  same  shaft ;  if  they  are  placed  on  the  same  line,  it  will  be 
impossible  to  drive  both  from  the  same  end,  and  as  this  is  important,  we 
move  the  third  row  the  width  of  the  pulley  5,  and  for  the  sake  of  unifor- 
mity, the  fourth  row  also.  Lay  off  now  the  length  of  looms  and  position 
of  pulleys  as  before,  and  parallel  with  the  pulleys  the  driving-pulleys  on 
the  shaft,  that  is  c  against  c,  f  against  f,  and  so  on.  Having  in  this  way 
plotted  in  all  the  looms,  every  alternate  set  being  on  a  line  with  the  third 
and  fourth  row,  we  proceed  now  to  lay  down  the  position  of  the  looms  in 
the  floor  above ;  and  since  for  economy  of  shafting  it  is  usual  to  drive 
from  the  lines  in  the  lower  rooms,  to  avoid  errors,  interference  of  belts  and 
pulleys,  it  is  usual  to  plot  the  upper  room  on  the  same  paper  or  board 
as  the  lower  room,  using  either  two  different  colored  inks,  or  drawing  the 
machines  in  one  room  in  deep  and  in  the  other  in  light  line,  as  shown  in 
plate  XXXY.  If  the  width  of  the  rooms  are  the  same,  the  lateral  lines 
of  looms  and  alleys  are  the  same,  and  it  is  only  necessary,  therefore, 
to  fix  the  end  lines.  Now,  as  the  first  loom  in  the  oiiter  row  of  looms, 
in  the  lower  room,  occupies  for  its  belt  the  position  k  on  the  shaft, 
the  loom  in  the  upper  room  must  be  moved  either  one  way  or  the 
other  to  avoid  this ;  thus  the  position  i  of  the  pulley  on  the  loom  must 
be  made  parallel  to  the  pulley  i  on  the  shaft,  so  in  the  other  looms  a  to  «, 
e  to  e,  d  to  d,  and  5  to  &. 

Besides  the  plan,  it  is  often  necessary,  and  always  conveniet,  to  draw  a 
sectional'  elevation  (as  in  fig.  283),  of  the  rooms,  with  the  relative  positions 
of  the  driving  pulleys  and  those  on  the  machines,  to  determine  suitably 
the  length  of  the  belts,  and  also  to  see  that  their  position  is  in  every  way 
the  most  convenient  possible.  For  instance,  in  the  figure,  one  of  the  lower 
belts  should  have  been  a  cross  belt,  and  one  of  the  upper  ones  straight : 
now  had  the  belts  to  the  second  row  of  looms  in  the  upper  story,  been 
drawn  as  they  should  have  been,  straight,  the  belt  would  have  interfered 
a  little  with  the  alley,  and  it  would  have  been  better  to  have  moved  the 
driving  shaft  a  trifle  towards  the  wall. 

From  this  illustration  of  the  location  of  machines,  knowing  all  the  re- 
quirements, in  a  similar  way  any  machinery  may  be  arranged  with  economy 
of  spaces,  materials,  power,  and  attendance.  These  two  last  items  are  of 
the  more  importance  as  they  involve  a  daily  expense,  where  the  others  are 
almost  entirely  the  first  outlay. 


DRAWING   OF   MACHINERY. 


201 


Fig.  2S4 


MAC  H  IN  E  S. 

Thus  far  the  illustrations  have  been  almost  entirely  confined  to  geomet- 
rical projections  and  the  delineation  of  parts  of  machinery.  We  now 
proceed  to  give  a  few  representations  of  complete  machines,  taken  from 
actual  constructions,  that  may  serve  not  only  as  copies  for  the  draughts- 
man, but  as  examples  for  the  engineer. 

In  many  cases,  where  the  mere 
working  of  the  machine  is  to  be 
shown,  a  few  lines  will  be  sufficient 
for  the  illustration,  as  in  fig.  284, 
which  is  a  skeleton  drawing  of 
Messrs.  Maudsley  and  Field's  direct 
acting  marine  engine,  in  which  A 
A1  are  the  steam  cylinders,  L  the  T 
piece  connecting  the  two  pistons,  E 
the  crank,  F  the  wheel  shaft,  and 
H  the  air  pump. 

Fig.  285  is  a  more  finished  draw- 
ing  of  the  same  machine,  in  which 
the  engine  frame  and  the  T  plate, 
with  its  connections  and  guides, 
are  shown.  It  is  very  common  in  drawings  to  express  piston  rods,  working 

beams,  and  cranks  thus,  by 
single  lines,  with  circles 
for  their  pins  or  pivots; 
as  their  forms  are  well 
established,"  and  their  full 
delineation  would  add  noth- 
ing to  the  information  to  be 
conveyed  to  the  mechanic, 
but  might  cover  up  and 
confuse  the  drawing  of 
more  important  parts  of  the 
machine. 

Figs.  286  and  287  are 
mere  outline  drawings  of 
two  English  locomotives, 
and  yet  sufficient  to  express 


Fig.  235. 


-202 


DRAWING    OF    MACHINERY. 


Fig.  2S6. 


the  form  of  the  engines,  and  the  arrangement  and  comparative  size  of 
drivers  and  other  wheels. 

"Working  drawings  are  complete  illustrations  of  a  machine,  either  as  a 
whole  or  in  detail,  sufficient  to  enable  the  mechanic  to  construct  it.  They 
should  be  of  a  large  enough  scale,  that  all  the  parts  may  be  readily  meas- 
ured, or  with  the  dimen- 
sions in  figures — this  last 
is  of  importance,  even 
when  the  scale  is  mea- 
surable. To  the  me- 
chanic, it  saves  time  and 
one  source  of  error,  but 
throws  more  responsi- 
bility on  the  draughts- 
man. Working  draw- 
~"  ings  should  be  almost 
entirely  in  line,  with  shading  only  sufficient  to  distinguish  circular  from 
flat  parts.  As  many  views  should  be  given  in  plans,  elevations,  sections, 
and  detailed  parts,  as  may  fully  explain  the  whole  construction  and  work- 
ing. In  the  designing  of  the  machine,  where  there  are  moving  parts,  it 
may  often  be  necessary 
to  draw  these  parts  in 
different  positions,  to  be 
sure  that  they  do  not  in- 
terfere with  some  other 
part  of  the  machine. 
If  these  lines  are  neces- 
sary to  the  mechanic, 
they  are  left  in;  one 
position  being  shown  in 
full  line,  the  other  or  others  in  dotted,  or  light,  or  red  lines. 

Figs.  288  and  289  are  sections,  at  right  angles  to  each  other,  of  the  cata- 
ract of  a  Cornish  Pumping  Engine.  To  make  these  complete  working 
drawings,  there  should  be  a  plan  in  addition  ;  but  if  it  be  explained  that 
all  the  parts  are  circular,  except  the  lever  e'  f  and  weight  A,  it  could  be 
readily  constructed,  especially  if  the  dimensions  were  figured.  In  explana- 
tion of  the  working  of  the  cataract,  it  may  be  briefly  said  to  consist  of  a 
cast-iron  cistern  G,  partly  filled  with  water,  in  which  is  a  pump  a',  the 
plunger  V  of  which  is  connected  with  a  lever  ef,  the  curved  end  of  which 
is  depressed  by  a  tappet  on  one  of  the  plug  rods  C,  and  the  plunger  thereby 


Fig.  2S7. 


DEAWESTG   OF   MACHINERY. 


203 


raised,  water  flowing  in  freely  through  the  valve  c' .  As  the  tappet  leaves 
the  lever,  the  plunger  is  forced  back  to  its  original  position  by  the  weight 
A,  and  with  speed  dependent  on  the  escape  of  water  from  beneath  the 
plunger  through  the  opening  left  round  the  regulating  plug  d,  the  descent 
of  the  plunger  detaches  catches  by  which  the  different  valves  of  the  steam 
cylinder  are  opened. 


Fig.  2SS. 


Plates  XXXVI.,  XXXYIL,  and  XXXVIII.  are  elevations,  sections, 
and  plans  of  the  48"  stop  gate  in  use  at  the  Nassau  Water  Works,  Brook- 
lyn, L.  I.,  forming  a  complete  set  of  working  drawings.  It  will  be  ob- 
served that  Plates  XXXYI.  and  XXXVII.,  which  are  views  at  right 
angles  to  each  other,  are  both  elevations  and  sections,  one-half  of  each 
being  in  elevation  and  one-half  in  section — a  method  in  use  to  economize 
the  number  of  drawings,  when  the  two  halves  are  complete  duplicates. 
Fig.  1,  PI.  XXXVIII.,  is  a  plan,  and  fig.  2  the  horizontal  section. 


204:  DRAWING    OF   MACHINERY. 

Plate  XXXIX.  fig.  1,  is  a  longitudinal  section  of  a  locomotive  boiler, 
and  fig.  2  an  interior  view  of  the  smoke  box. 

The  steam  space  C  at  the  fire  end  of  the  boiler  is  half  globe  shaped, 
and  surmounted  by  a  dome.  The  object  of  the  dome  is  to  carry  the  steam 
as  high  as  possible  above  the  water  line  before  its  introduction  into  the 
steam  pipe  p,  in  order  that  the  water  held  in  suspension  near  the  surface  of 
the  water,  may  not  be  carried  over  into  the  cylinders.  The  steam  pipe  trav- 
erses the  length  of  the  boiler,  and  in  the  smoke  box  branches  off  to  each 
cylinder  ;  I  is  the  regulating  or  throttle  valve,  worked  by  the  handle  which 
passes  out  through  a  stuffing  box  in  the  end  of  the  boiler ;  j  is  the  fire  box. 
surrounded  on  all  sides  except  at  bottom  with  a  water  space ;  the  top  or 
crown  sheet  of  the  fire  box  is  strengthened  by  pieces  of  iron,  and  the  flat 
sides  are  securely  bolted  together.  The  tube  sheet  is  sufficiently  stayed 
by  the  tubes  themselves ;  this  sheet  is  often  made  of  copper,  as  are  the 
side  sheets,  from  6  to  7  inches  below,  to  about  the  same  height  above  the 
coal  line,  in  locomotive  boilers  burning  anthracite  or  bituminous  coal ;  -7? 
are  the  fire  tubes  varying  in  different  boilers  from  li  to  3  inches  in  diam- 
eter and  from  T  to  14  feet  in  length;  the  longer  the  tube  the  larger  the 
diameter.  II II  are  the  cylinders,  a?  a?  the  steam  chests  ;  uu  the  exhaust 
pipes,  which  are  connected  together,  and  pass  up  into  the  centre  of  the 
smoke  stack  or  chimney.  The  exhaust  furnishes  by  its  blast  draught  to 
the  chimney,  and  if  the  outlet  be  contracted,  the  greater  the  force  of  the 
issuing  steam,  and  the  stronger  the  draught ;  but  of  course  the  greater  the 
back  pressure  in  the  cylinders. 

Plate  XL.  is  the  front  elevation,  and  Plate  XLI.  is  the  side  ele- 
vation and  section  through  air-pump  of  one  of  the  oscillating  engines  of  the 
Golden  Gate,  in  which  a  is  the  main  shaft,  5  crank-pin,  c  cylinder,  d  trun- 
nions on  which  the  cylinder  oscillates  to  accommodate  itself  to  the  motion 
of  the  crank.  0,  stuffing-box  on  the  cylinder  head.  This  is  made  as  long 
as  practicable,  to  give  as  much  bearing  as  possible  for  oscillating  the 
cylinder,  ff,  belt-passage  connecting  the  trunnion  with  g  g,  side  pipe. 
h  h,  valve-stems  connecting  with  the  balance  puppet- valves,  in  i  i  valve- 
chests.  The  lower  valve  on  the  right  or  steam  side  is  concealed  by  j, 
air-pump.  The  air-pump  bucket  is  provided  with  India-rubber  valves,  and 
is  worked  by  A*,  crank  on  the  intermediate  shaft.  I  Z,  condenser.  There 
are  two  condensers  and  two  air-pumps,  they  are  located  between  the  cylin- 
ders and  inclined  towards  each  other,  one  only  being  represented. 

The  passage  ff,  together  with  the  side  pipes,  valve-chests,  and  appur- 
tenances, are  fixed  to  the  cylinder,  and  oscillate  with  it,  the  steam  being 
received  through  one  trunnion,  and  allowed  to  escape  to  the  condenser 


DRAWING   OF   MACHINERY.  205 

through  the  opposite  one.  m  is  an  injection  cock,  admitting  the  water 
upon  a  scattering  plate  in  the  condenser. 

The  valves  are  worked  by  the  toes  o  o  in  the  usual  manner.  The  rock- 
shafts  pp  receive  motion  partly  from  the  movement  of  the  cylinder,  and 
partly  from  the  eccentric.  Levers  are  permanently  attached  to  the  trip- 
shafts  q  <?,  the  ends  of  which  work  in  a  slotted  piece  curved  to  the  centre 
of  the  trunnion.  This  piece  is  guided,  as  represented  in  the  engraving,  by 
vertical  rods  sliding  in  bushes  attached  to  the  fixed  framing,  and  is  con- 
nected by  a  rod  to  the  starting-lever  r  /  all  the  levers  for  working  by  hand 
being  so  balanced,  that  the  engineer  with  one  hand  can  work  the  engine 
up  to  the  usual  speed. 

The  cut-off  valve  is  placed  outside  the  trunnion,  and  is  a  balance  pup- 
pet-valve, worked  by  the  ordinary  cam  motion,  and  so  arranged  as  to  act 
either  as  cut-off  or  throttle,  or  both,  the  levers  being  placed  within  reach 
of  the  engineer  when  working  the  engine. 

Plate  XL1I.  is  a  vertical  section  through  the  centre  of  a  TURBINE 
"WHEEL,  and  the  axis  of  the  supply  pipe.*  Plate  XLIII.  is  a  plan  of  the 
Turbine  and  wheelpit.  Fig.  1,  Plate  XLIY.,  is  a  plan  of  the  whole  wheel, 
the  guides  and  garniture.  This  Turbine  was  constructed  for  the  Tremont 
Manufacturing.  Co.  at  Lowell,  by  Mr.  James  B.  Francis,  and  contains 
most  of  Mr.  Boyden's  improvements.  Its  expenditure  of  water  under  13 
feet  head  and  fall,  is  about  139  cubic  feet  per  second,  and  its  ratio  of  useful 
effect  to  the  power  expended,  about  79  per  cent. 

B,  the  surface  of  the  water  in  the  wheelpit,  represented  at  the  lowest 
height  at  which  the  turbine  is  intended  to  operate.  C,  the  masonry  of  the 
wheelpit.  D,  the  floor  of  the  wheelpit.  To  resist  the  great  upward  pres- 
sure which  takes  place  when  the  wheelpit  is  kept  dry  by  pumps,  three  cast- 
iron  beams  are  placed  across  the  pit,  the  ends  extending  about  a  foot  under 
the  walls  on  each  side ;  on  these  are  laid  thick  planks,  which  are  firmly 
secured  to  the  cast-iron  beams  by  bolts.  To  protect  the  thick  planking 
from  being  worn  out  by  the  constant  action  of  the  water,  they  are  covered 
with  a  flooring  of  one  inch  boards.  E,  the  wrought-iron  supply  pipe. 
This  is  constructed  of  plate  iron  three-eighths  of  an  inch  thick,  riveted 
together.  The  supply  pipe  is  furnished  with  the  man  hole  and  ventilating 
pipe  G-,  and  the  leak  box  H,  to  catch  the  leakage  of  the  head  gate,  when- 
ever it  is  closed  for  repairs  of  the  wheel. 

The  lower  end  of  the  supply  pipe  is  formed  by  the  cast-iron  curbs  III. 
The  curbs  are  supported  from  the  wheelpit  floor  by  four  columns,  resting 

*  By  permission  of  the  author  we  take  the  following  plates  and  description  from  the  standard 
work,  "  Lowell  Hydraulic  Experiments." 


206  DRAWING    OF    MACHINERY. 

on  the  cast-iron  beam,  O ;  the  beams  X',  rest  immediately  upon  the  col- 
umns, and  the  curb  upon  the  beams,  the  latter  projecting  over  the  columns 
far  enough  for  that  purpose.  The  beams  ~N'  also  act  as  braces  from  the 
wheelpit  wall  to  the  curb,  and  are  strongly  bolted  at  each  end. 

K,  the  disc.  This  is  of  cast-iron,  and  is  turned  smooth  on  the  upper 
surface,  and  also  on  its  circumference.  It  is  suspended  from  the  upper 
curb  I,  by  means  of  the  disc  pipes  31 M.  The  disc  carries  on  its  upper 
surface,  thirty-three  guides  (fig.  1,  Plate  XLIV.,)  for  the  purpose  of  giving 
the  water  entering  the  wheel,  proper  direction.  They  are  made  of  Rus- 
sian plate  iron,  one-tenth  of  an  inch  in  thickness,  secured  to  the  disc  by 
tenons,  riveted  on  the  under  side.  The  upper  corners  of  the  guides,  near 
the  wheel,  are  connected  by  the  garniture  L,  which  is  intended  to  diminish 
the  contraction  of  the  streams  entering  the  wheel,  when  the  regulating 
gate  is  fully  raised.  The  garniture  is  composed  of  thirty-three  pieces  of 
cast-iron,  carefully  fitted  to  fill  the  spaces  between  the  guides ;  they  are 
strongly  riveted  to  the  guides  and  to  each  other. 

The  upper  flange  of  the  disc  pipe  is  furnished  with  adjusting  screws, 
by  which  the  weight  is  supported  upon  the  upper  curb.  The  escape  of 
water  between  the  upper  curb  and  the  upper  flange  of  the  disc  pipe,  is 
prevented  by  a  band  of  leather  on  the  outside,  which  is  retained  in  its 
place  by  the  wrought-iron  ring.  The  top  of  the  disc  pipe,  just  below 
the  upper  flange,  has  two  wings,  fitting  into  recesses  in  the  top  of  the  curb, 
to  prevent  the  disc  from  rotating  in  the  opposite  direction  to  the  wheel. 

R,  H,  the  regulating  gate.  [Represented,  Plate  XLII.,  as  fully  raised. 
The  gate  is  of  cast-iron ;  the  upper  part  of  the  cylinder  is  stiffened  by  a 
rib,  to  which  are  attached  three  brackets,  S,  S.  To  these  brackets  are  at- 
tached wrought-iron  rods,  by  which  the  gate  is  raised  or  lowered.  To  one 
of  the  rods  is  attached  the  rack  Y.  The  other  two  rods  are  attached  by 
means  of  links,  to  the  levers  T  T.  The  other  ends  of  these  levers  carry 
geered  arch  heads,  into  which,  and  into  the  rack  Y,  work  three  pinions, 
"W,  of  equal  pitch  and  size,  fastened  to  the  same  shaft,  so  arranged  that  by 
the  revolution  of  the  pinion  shaft,  the  gate  is  moved  up  or  down,  equally 
on  all  sides.  The  shaft  on  which  the  pinions  are  fastened,  is  driven  by  the 
worm  wheel  X ;  this  is  driven  by  the  worm  a,  either  by  the  governor  Y, 
or  the  hand  wheel  Z.  The  shaft  on  which  the  worm  a  is  fastened,  is  fur- 
nished with  movable  couplings,  which,  when  the  speed  gate  is  at  any  inter- 
mediate points  between  its  highest  aud  lowest  positions,  are  retained  in 
place  by  spiral  springs  ;  in  either  of  the  extreme  positions,  the  couplings 
are  separated  by  means  of  a  lever  moved  by  pins  in  the  rack  Y ;  by  this 
means,  both  the  regulator  and  hand  wheel  are  prevented  from  moving  the 


DRAWING    OF   MACHINERY.  207 

gate  in  one  direction,  when  the  gate  has  attained  either  extreme  position. 
If,  however,  the  regulator  or  hand  wheel  should  be  moved  in  the  opposite 
direction,  the  couplings  would  catch,  and  the  gate  would  be  moved.  The 
weight  of  the  gate  is  counterbalanced  by  weights  attached  to  the  levers 
T  T,  and  by  the  intervention  of  a  lever  to  the  rack  V. 

l>  &,  the  wheel,  consists  of  a  central  plate  of  cast-iron,  and  two  crowns, 
c  c,  of  the  same  material  to  which  the  buckets  are  attached.  The  buckets 
are  forty-four  in  number,  made  of  Eussian  plate  iron,  T»¥  of  an  inch  in 
thickness,  and  are  secured  to  the  crowns  by  grooves  cut  in  the  crowns  of 
the  exact  form  of  the  buckets,  and  by  tenons  entered  into  the  mortises 
in  both  crowns,  and  riveted  on  the  opposite  sides. 

dd,  the  vertical  shaft,  of  wrought-iron,  runs  upon  a  series  of  collars, 
resting  upon  corresponding  projections  in  the  suspension  box  e'.  The  part 
of  the  shaft  on  which  the  collars  are  placed,  is  made  separate  from  the 
main  shaft,  and  is  pinned  to  it  at/",  by  means  of  a  socket  in  the  top  of  the 
main  shaft,  which  receives  a  corresponding  part  of  the  collar  piece.  The 
collars  are  made  of  cast  steel ;  they  are  separately  screwed  on,  and  keyed 
to  a  wrought-iron  spindle. 

The  suspension  box  is  made  in  two  parts,  to  admit  of  its  being  taken 
off  and  put  on  the  shaft ;  it  is  lined  with  Babbit  rrfetal.  It  is  found  that 
bearings  thus  lined  will  carry  from  fifty  to  a  hundred  pounds  to  the  square 
inch,  with  every  appearance  of  durability. 

f'f,  the  upper  and  lower  bearings  are  of  cast-iron,  lined  with  Babbit 
metal,  adjustable  horizontally  by  means  of  screws.  The  suspension  box  <?', 
rests  upon  the  gimbal  g.  The  gimbal  itself  is  supported  on  the  frame  h  h 
by  adjusting  screws,  which  give  the  means  of  raising  and  lowering  the 
suspension  box,  and  with  it,  the  vertical  shaft  and  wheel.  The  lower  end 
of  the  shaft  is  fitted  with  a  cast-steel  pin,  i.  This  is  retained  in  its  place 
by  the  step,  which  is  made  in  three  parts,  and  lined  with  case-hardened 
wrought-iron. 

The  weight  of  the  wheel,  upright  shaft,  and  bevel  geer,  is  supported 
by  means  of  the  suspension  box  e'  on  the  frame  &,  which  rests  upon  the 
long  beams  m,  reaching  across  the  wheelpit,  and  supported  at  the  ends  by 
the  masonry,  and  also  at  intermediate  points  by  the  braces  n  n. 

Mr.  Francis  deduces  the  following  rules  for  proportioning  turbines : 

The  sum  of  the  shortest  distances  between  the  buckets,  should  be  equal  to  the  diameter 
of  the  wheel. 

The  height  of  the  orifices  at  the  circumference  of  the  wheel,  should  be  equal  to  one- 
tenth  of  the  diameter  of  the  wheel. 

The  width  of  the  crowns  should  be  four  times  the  shortest  distance  between  the  buckets. 


208  DRAWING   OF   MACHINERY. 

« 

The  sum  of  the  shortest  distances  between  the  curved  guides,  taken  near  the  wheel, 
should  be  equal  to  the  interior  diameter  of  the  wheel. 

The  number  of  buckets  is,  to  a  certain  extent,  arbitrary.  As  a  guide  in  practice,  to 
be  controlled  by  particular  circumstances,  and  limited  to  diameters  of  not  less  than  two 
feet,  the  number  of  buckets  should  be  three  times  the  diameter  in  feet,  plus  thirty.  The 
Tremont  Turbine  is  8^  feet  in  diameter,  and  according  to  the  proposed  rule,  should  have 
fifty-five  buckets  instead  of  forty-four.  The  number  of  the  guides  is  also  to  a  certain  ex- 
tent arbitrary ;  the  practice  at  Lowell  has  been,  usually,  to  have  from  a  half  to  three- 
fourths  of  the  number  of  buckets. 

As  turbines  are  generally  used,  a  velocity  of  the  interior  circumference  of  the  wheel, 
of  about  fifty-six  per  cent,  of  that  due  to  the  fall  acting  upon  the  wheel,  appears  most 
suitable. 

To  lay  out  the  curve  of  the  buckets. 

Referring  to  Plate  XLIV.,  fig.  2,  the  number  of  buckets,  j\7J  having  been  determined 

by  the  preceding  rules,  set  off  the  arc  g  i  =  -^-=-  .  Let  o>  =  g  A,  the  shortest  distance 
between  the  buckets :  t  the  thickness  of  the  metal  forming  the  buckets.  Make  the  arc 
g  Te  =  5a>.  Draw  the  radius  Ok,  intersecting  the  interior  circumference  of  the  wheel  at  I; 
the  point  I  will  be  the  inner  extremity  of  the  bucket.  Draw  the  directrix  Z  TO  tangent  to 
the  inner  circumference  of  the  wheel.  Draw  the  arc  o  n,  with  the  radius  o>  -f  t,  from  i,  as 
a  centre ;  the  other  directrix  gp,  must  be  found  by  trial,  the  required  conditions  being, 
that,  when  the 'line  m  Us  revolved  round  to  the  position  g  t,  the  point  TO  being  constantly 
on  the  directrix  gp,  and  another  point  at  the  distance  mg  =  rs,  from  the  extremity  of  the 
line  describing  the  bucket,  being  constantly  on  the  directrix  m  I,  the  curve  described  shall 
just  touch  the  arc  no.  A  convenient  line  for  a  first  approximation,  may  be  drawn  by 
making  the  angle  0  g  p  =  11°.  After  determining  the  directrix  according  to  the  preceding 
method,  if  the  angle  Ogp  should  be  greater  than  12°,  or  less  than  10°,  the  length  of  the 
arc  g'Tc  should  be  changed,  to  bring  the  angle  within  these  limits. 

The  trace  adopted  for  the  corresponding  guides  is  as  follows : — The  number  n  having 
been  determined,  divide  the  circle  in  which  the  extremities  of  the  guides  are  found,  into  n 
equal  parts,  v  w,  w  a-,  &c.  Put  a>'  for  the  width  between  two  adjoining  guides,  and  t'  for 

the  thickness  of  the  metal  forming  the  guides.  AYe  have  by  rule,  a>'  =•  — .  With  w'  as 
a  centre,  and  the  radius  <a'  +  t\  draw  the  arc  yz;  and  with  a  as  a  centre,  and  the  radius 
2(«'  +  £'),  draw  the  arc  a'  &'.  Through  v  draw  the  portion  of  a  circle  v  c',  touching  the 
arcs  y  z  and  a!  V;  this  will  be  the  curve  for  the  essential  part  of  the  guide.  The  remainder 
of  the  guide,  c'  d\  should  be  drawn  tangent  to  the  curve  c'  v  ;  a  convenient  radius  is  one 
that  would  cause  the  curve  c'  d\  if  continued,  to  pass  through  the  centre  0. 


ARCHITECTURAL   DRAWING.  209 


AKCHITECTUKAL  DKAWING. 

THE  art  of  architecture  consists  in  the  designing  of  a  building,  so  as  to 
be  most  suitable  and  convenient  for  the  purposes  for  which  it  is  intended ; 
in  selecting  and  disposing  of  the  materials  of  which  it  is  composed,  so  as  to 
withstand  securely  and  permanently  the  strains  and  wear  to  which  they 
may  be  subjected ;  and  arranging  the  parts  so  as  to  produce  the  most  artis- 
tic effect  consistent  with  the  use  of  the  building  and  its  location,  and  ap- 
plying to  it  such  appropriate  ornament  as  may  express  the  purpose,  and 
harmonize  with  the  construction. 

As  an  art  it  consists  in  the  convenient  and  appropriate  combination  of 
established  forms  and  ornamentations,  and  the  adaptation  of  old  and  recog- 
nized styles  to  present  requirements.  Success  in  its  practice  should  there- 
fore depend  not  only  on  the  talent  of  the  architect,  but  on  a  thorough  study 
of  the  best  masters  of  the  science,  and  an  extended  acquaintance  with  the 
effects  of  practical  examples  and  the  principles  of  construction. 

In  domestic  architecture,  by  far  the  most  extensive  branch  of  the  pro- 
fession, most  persons  can  give  some  idea  of  the  kind  of  building  which  they 
wish  to  have  constructed,  and  perhaps  express  by  line  the  general  arrange- 
ment of  rooms ;  but  it  is  left  to  the  architect  to  settle  the  style  of  building 
appropriate  to  the  position,  to  determine  the  dimensions  and  positions  of 
rooms  and  passages,  thickness  of  walls  and  partitions,  arrangements  for 
drainage,  heating,  and  ventilating — in  fact  all  the  details  which  are  to  be 
drawn  or  specified  for  the  construction.  It  is,  therefore,  indispensable  that 
he  should  understand  what  are  the  best  proportions  of  parts,  and  what  are 
the  necessities  of  construction. 


FOUNDATIONS. 

In  preparing  for  the  foundation  of  any  building,  there  are  two  sources 
of  failure  to  be  guarded  against — inequality  of  settlement,  and  lateral  escape 


210  ARCHITECTURAL   DRAWING. 

of  the  supporting  material.  It  is  not  so  much  an  unyielding  as  a  uni- 
formly yielding  foundation  that  is  required.  When  the  character  of  the 
ground  is  not  known,  it  is,  therefore,  important  that,  previous  to  the  com- 
mencement of  the  work,  soundings  should  be  taken  to  ascertain  the  nature 
of  the  soil  and  the  lay  of  the  strata,  to  determine  the  kind  of  foundation ; 
and  the  more  important  and  weighty  the  superstructure,  the  more  careful 
and  deeper  the  examination. 

Natural  foundations. — The  best  foundation  is  a  natural  one,  such  as  a 
stratum  of  rock  or  compact  gravel.  If  circumstances  prevent  the  work 
being  commenced  from  the  same  level  throughout,  the  ground  must  be 
carefully  benched  out,  i.  e.,  cut  into  horizontal  steps,  so  that  the  courses 
may  all  be  perfectly  level.  It  must  also  be  borne  in  mind,  that  all  work 
will  settle  more  or  less,  according  to  the  perfection  of  the  joints,  and 
therefore  in  these  cases  it  is  best  to  bring  up  the  foundations  to  a  uniform 
level  with  large  blocks  of  stone  or  with  concrete,  before  commencing  the 
superstructure,  which  would  otherwise  settle  most  over  .the  deepest  parts, 
on  account  of  the  greater  number  of  mortar  joints,  and  thus  cause  unsightly 
fractures.  Foundations  in  soil  should  be  excavated  to  a  depth  below  the 
action  of  frost. 

Artificial  foundations. — "Where  the  ground  in  its  natural  state  is  too 
soft  to  bear  the  weight  of  the  proposed  structure,  recourse  must  be  had  to 
artificial  means  of  support,  and,  in  doing  this,  whatever  mode  of  construc- 
tion be  adopted,  the  principle  must  always  be  that  of  extending  the  bear- 
ing surface  as  much  as  possible.  There  are  many  ways  of  doing  this — as 
by  a  thick  layer  of  concrete  or  beton,  or  by  layers  of  planking,  or  by  a  net- 
work of  timber,  or  by  increasing  width  of  wall,  or  these  different  methods 
may  be'  combined.  The  weight  may  also  be  distributed  over  the  entire 
area  of  the  foundation  by  inverted  arches.  The  use  of  timber  is  objection- 
able where  it  cannot  be  kept  constantly  wet,  as  alternations  of  dryness 
and  moisture  soon  cause  it  to  rot,  and  for  such  localities  concrete  is  to  be 
preferred. 

To  prevent  the  lateral  escape  of  the  supporting  material,  when  build- 
ing in  running  sand  or  soft  clay,  which  would  ooze  out  from  below  the 
work  and  allow  the  superstructure  to  sink,  in  addition  to  protecting  the 
surface  with  planking,  concrete,  or  timber,  it  is  often  necessary  to  enclose 
the  whole  area  of  the  foundation  with  piles  of  timber  or  plank  driven  close 
together ;  this  is  called  sheet-piling. 

Where  there  is  a  hard  stratum  below  the  soft  ground,  but  at  too  great 
a  depth  to  allow  of  the  solid  work  being  brought  up  from  it  without  greater 
expense  than  the  circumstances  of  the  case  will  allow,  it  is  usual  to  drive 


ARCHITECTURAL   DRAWING. 


211 


down  wooden  piles,  often  shod  with  iron,  until  their  bottoms  are  firmly 
fixed  in  the  hard  ground.  The  upper  ends  of  the  piles  are  then  cut  off 
level,  and  covered  with  a  platform  of  timber,  on  which  the  work  is  built 
in  the  usual  way.  The  piles  are  generally  of  about  1  foot  diameter,  and 
are  driven  at  distances  of  from  2  to  3  feet  from  centre  to  centre. 

Where  a  firm  foundation  is  required  to  be  formed  in  a  situation  where 
no  firm  bottom  can  be  found  within  an  available  depth,  piles  are  driven, 
to  consolidate  the  mass,  a  few  feet  apart  over  the  whole  area  of  the  foun- 
dation, which  is  surrounded  by  a  row  of  sheet-piling  to  prevent  the  escape 
of  the  soil ;  the  space  between  the  pile-heads  is  then  filled  to  the  depth  of 
several  feet  with  stones  or  concrete ;  and  the  whole  is  covered  with  a  timber 
platform  on  which  to  commence  the  solid  work. 

WALLS. 

"Walls  of  permanent  structures  are  almost  exclusively  composed  of 
either  stone  or  brick,  or  both,  and  are  included  in  one  general  term  as 
masonry. 

Fig.  1  represents  the  front  of  a  wall  called  the  face ;  fig.  2,  a  section ; 
and  fig.  3,  the  view  of  rear,  or  the  "backing.  The  interior  of  the  wall  is 


1  „„ 

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Fig.  1. 


Fig.  2. 


JLJL 


Fig.  3. 


called  the  filling.  The  term  course  is  applied  to  each  horizontal  layer  of 
stone  or  brick ;  if  all  the  stones  in  a  layer  are  of  equal  thickness,  it  is 
termed  regular  coursing,  footings  are  the  lower  projecting  courses  (figs. 
1  and  2)  resting  on  the  foundation,  usually  not  less  than  double  the  width 
of  the  wall  above  in  walls  of  buildings ;  but  for  other  walls,  the  width  de- 
pends on  the  nature  of  the  foundation.  Hock  foundations  need  no  extra 
width  of  wall.  String  courses,  or  belting,  are  upper  courses  projecting 


212  AECeiTECTUEAL  DKAWIXG. 

slightly  beyond  the  face  of  the  wall.  Coping  is  the  top  courses,  usually 
got  out  in  considerable  lengths  in  comparison  with  the  stones  in  the  rest 
of  the  work. 

The  beds  of  a  stone  or  brick  are  the  surfaces  on  which  they  rest ;  the 
build  is  the  upper  bed  on  which  the  stone  above  is  placed ;  the  inter- 
stices between  the  stones  are  termed  joints.  /Stretchers  are  stones  or  brick 
which  have  their  length  disposed  lengthways  of  the  wall ;  headers  have 
their  length  crossways.  Quoins  are  the  corners  of  a  wall.  Bond  is  the 
lapping  of  the  stones  or  brick  on  each  other  in  the  ^  construction,  so  as  to- 
tie  the  separate  pieces  together.  Three  classes  of  bond  are  shown  in  the 
face  (fig.  1) ;  the  lowest  six  courses  consist  of  alternate  courses  of  headers 
and  stretchers,  the  next  six  courses  above  have  alternate  headers  and 
stretchers  in  the  same  course,  and  in  the  remaining  courses  a  header  occurs 
at  every  third  stone ;  this  is  the  most  usual  bond.  Headers  should  not  be 
placed  one  above  the  other  in  alternate  courses. 

Figs.  4  and  5  represent  brick  bonds ; — fig.  4,  the  old  English  bond,  and 


,.  >l      f      t      i      I      1 

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lilt 

1 

Fig.  4.  Fig.  5. 

fig.  5,  the  Flemish  bond.  The  most  common  bond  in  this  country  is  to 
lay  a  certain  number  of  stretching  courses,  and  then  a  heading  course. 
The  fire-law  of  New  York  requires  brick  work  to  be  built  with  headers 
every  five  courses,  but  every  seventh  course  a  heading  course  is  more  com- 
monly used.  In  all  masonry,  no  vertical  joint  should  extend  through  two 
courses,  but  the  vertical  joints  should  be  as  near  as  possible  midway  be- 
tween those  below ;  in  other  words,  break  joint  with  them. 

Walls  are  composed  of  stones  laid  either  with  or  without  mortar.  The 
latter  is  called  dry  masonry ;  rough  wall  is  dry  work  of  rough  stones  ;  if 
laid  in  mortar,  it  is  called  rubble  work,  but  frequently  this  term  is  made  to 
include  all  rough  work.  Cut  stone  is  called  ashler  /  if  only  cut  on  beds 
and  joints,  it  is  called  rock  or  quarry  faced  ashler. 

On  the  thickness  of  walls. — Retaining-^dhs  are  such  as  sustain  a  lateral 
pressure  from  an  embankment  or  head  of  water  (figs.  6  and  7).  The  width 
of  a  retaining-wall  depends  upon  the  height  of  the  embankment  which  it 
may  have  to  sustain,  and  the  kind  of  earth  of  which  it  is  composed  (the 
steeper  the  natural  slope  at  which  the  earth  would  stand,  the  less  the  thrust 


AKCHTTECTTJKAL   DRAWING. 


213 


Fig.  6. 


against  the  wall),  and  the  comparative  weight  of  the  earth  and  of  the  ma- 
sonry. The  formula  given  by  Morin  for  ordinary  earths  and  masonry  is 
I  =  0.285  h  +  h' ;  that  is,  to 
find  the  breadth  of  a  wall 
laid  in  mortar,  multiply  the 
whole  height  of  the  embank- 
ment above  the  footing  by 
285 
1000 
thickness  one-fourth  more. 

Most  retaining  or  brick 
walls  have  an  inclination  or 
batter  to  the  face,  sometimes 
also  the  same  in  the  back,  but  offsets  (fig.  6)  are  more  common.  The  usual 
batter  is  from  1  to  3  inches  horizontal  for  each  foot  vertical.  To  determine 
the  thickness  of  a  wall  having  a  batter,  "  determine  the  width  by  the  rule 
above,  and  make  this  width  at  one-ninth  of  the  height  above  the  base." 

In  most  large  cities  there  are  Building  Acts  in  force,  to  which  all  con- 
structions within  their  limits  must  conform.  Copious  extracts  from  the 
New  York  law  may  be  found  in  the  Appendix. 

Figs.  8,  9,  10,  11,  p.  214,  represent  the  thickness  of  external  brick  walls 
to  the  first,  second,  third,  and  fourth-rate  buildings,  as  provided  by  the 
Building  Act  for  the  city  of  London.  Figs.  12, 13, 14, 15,  show  the  same  with 
respect  to  party-walls.  The  figures  1,  If,  2,  2f,  represent  the  number  of 
lengths  of  brick  in  the  wall.  The  following  table  gives  the  rate  in  Liver- 
pool, not  differing,  constructively,  materially  from  that  of  London. 


HEIGHTS  AKD  WIDTHS  OF  BUILDINGS. 

First-rate  dwelling-house. 

Second-rate  dwelling-house. 

Third-rate  dwelling-house. 

Fonrth-rate  dwelling-house. 

Exceeding  forty  -four  feet 
in  height,  or  twenty  -seven 
feet  front. 

Not  exceeding  forty-four 
feet  in  height,  or  twenty- 
seven  feet  front 

Not  exceeding  thirty-six 
feet  in  height,  or  twenty- 
one  feet  front. 

Not  exceeding  thirty-two 
feet  in  height,  or  fifteen 
feet  front. 

"  Every  brewery,  distillery,  manufactory,  or  warehouse,  of  whatever 
height  or  extent  of  frontage,  is  considered  to  be  a  first-rate  building,  the 
external  walls  of  which  are  in  their  respective  stories  to  be  2f ,  2,  and  1£ 
bricks  in  thickness,  and  the  party-walls  of  2  and  1|  bricks." 

Walls  built  of  rubble  should  also  be  somewhat  thicker  than  those  of 
brick — on  an  average  at  least  one  quarter  thicker,  but  depending  on  the 
character  of  the  stone.  The  common  form  of  cut- work  is  to  make  the  face 
ashler,  and  back  up  with  rubble  or  brick. 


ARCHITECTURAL    DRAWING. 


Fig.  12 


Fig.  8 


Fig.  10  Fig.  9 


Fig.  11 


Fig.  15 


X 


rnLirnrj 


dJL~l 


a  an  an  ±11 


I  U\  I   2 JL  2 

CZl  OUEZJLlJLJrillUJ 


81 


ARCHITECTURAL    DRAWING. 


215 


Mortar  is  a  mixture  of  lime  or  cement,  or  both,  with  water  and  sand. 
In  the  preparation  of  mortar,  the  materials  should  be  well  selected ;  the 
sand  sharp  and  clean,  the  proportions  properly  preserved,  and  the  whole 
intimately  mixed.  As  a  general  rule,  the  lime  or  cement  should  be  suf- 
ficiently fine  to  cover  all  the  grains  of  sand,  with  the  thinnest  possible  stra- 
tum. Practically,  about  three  or  four  cubic  feet  of  sand  are  added  to  one 
cubic  foot  of  half  liquid  lime,  for  fat  limes ;  lean  lime  may  not  bear  more 
than  half  this  sand.  Cement  is  generally  mixed  with  sand  in  proportions  of 
1  to  3,  but  in  situations  where  a  quick  set  is  necessary,  in  equal  proportions. 

Arches. — Arches  are  of  various  shapes,  as, 


.i^  ^~ 
\ 

/ 

\          / 

/ 

V 

Elliptical, 

Segmental, 

Circular. 

Pointed, 


The  outer  surface  of  the  arch  is  called  the  extrados  or  back  of  the  arch, 
the  inner  or  concave  surface  the  intmdos  or  the  soffit;  the  joints  of  all 
arches  should  be  perpendicular  to  the  surface  of  the  soffit.  The  stones 
are  called  arch  stones  or  voussoirs.  The  first  course  on  each  side  are 
termed  springers,  which  rest  on  the  imposts  or  abutments.  In  case  of  a 
segmental  arch,  the  course  beneath  the  springers  are  called  skew-backs. 
The  extreme  width  between  springers  is  called  the  span  of  the  arch,  and 
the  versed  sine  of  the  curve  of  the  soffit  the  rise  of  the  arch.  The  highest 
portion  of  the  arch  is  called  the  crown,  and  the  centre  course  of  voussoirs 
the  key-course.  The  side  portions  of  arches  between  the  springing  and  the 
crown,  are  termed  haunches  or  flanks.  All  arches  should  be  well  sustained 
by  backing  on  the  haunches,  called  spandrel-backing.  The  line  of  inter- 
section of  arches  cutting  across  each  other  transversely  is  called  a  groin, 
and  the  arches  themselves  groined  arches. 

That  the  voussoirs  of  an  arch  may  resist  crushing,  they  must  have  a 
certain  depth  proportioned  to  the  pressure  of  the  arch ;  and  as  this  in- 
creases from  the  curve  towards  the  springing,  the  depth  of  the  voussoirs 
should  likewise  increase  from  the  crown  to  the  springing.  Peronnet  has 
given  as  a  rule  for  the  depth  at  the  crown  the  formula  d  =  .07  r  +  1  foot, 
in  which  formula  r  is  the  greatest  radius  of  curvature  of  the  in  trades. 
This  formula  is  applicable  to  arches  less  than  fifty  feet  radius ;  but  beyond 
this  it  gives  greater  dimensions  than  in  ordinary  practice. 


216 


ARCHITECTURAL   DRAWING. 


FRAMING. 

Framing  is  the  art  of  arranging  beams  for  the  various  purposes  to 
which  they  are  applied  in  structures.  Timber  and  iron  are  the  only  ma- 
terials in  common  use  for  frames. 

"Wooden  beams  are  usually  represented  longitudinally  both  in  eleva- 
tion and  in  section  by  their  outlines  merely,  or  in  end  view  by  rectangles 
with  diagonals  from  opposite  corners  (fig.  18),  and  in  end  section  by  the 

usual  diagonal  lines  in 


one  direction  across  the 
face.  Sometimes  in 
more  finished  drawings, 
or  when  a  distinction  is  to  be  marked  between  different  materials,  the 
grain  of  the  wood  is  represented  as  in  figs.  19  and  20,  a  side  and  end  ele- 
vation of  a  beam. 

Flooring. — The  timbers  Avhich  support  the  flooring-boards  and  ceiling 
of  a  room  are  called  the  naked  flooring. 

The  simplest  form  of  flooring,  and  the  one  usually  adopted  in  the  con- 
struction of  city  houses  and  stores  is  represented  in  plan  and  section,  fig. 
21.  It  consists  of  a  single  series  of  beams  or  deep  joists,  reaching  from 
wall  to  wall.  As  a  lateral  brace  between  each  set  of  beams,  a  system  of 


Fig.  13. 


Fig.  19. 


Fig.  20. 


11 


Fig.  21. 


bridging  is  adopted,  of  which  the  best  is  the  herring-bone  bridging,  formed 


ARCHITECTURAL   DRAWING. 


217 


of  short  pieces  of  joists  about  2x3,  crossing  each  other,  and  nailed  securely 
to  the  top  and  bottoms  of  the  several  beams,  represented  by  a  and  5  in 
fig.  21,  and  wherever  a  flue  occurs,  or  a  stairway  or  well-hole  prevents  one 
or  more  joists  from  resting  on  the  wall,  a  header,  H,  is  framed  across  the 
space  into  the  outer  beams  or  trimmer-beams  T  T,  and  the  beams  cut  off 
or  tail-beams  are  framed  into  the  trimmer. 

Whenever  the  distances  between  the  walls  exceed  the  length  that  can 
safely  be  given  to  joists  in  one  piece,  an  intermediate  beam  or  girder, 
running  longitudinally,  is  introduced,  into 
which  the  joists  are  framed  (fig.  22).  Very 
often  the  joists  are  merely  notched  on  to 
beams.  Flooring  is  still  further  varied,  by 
framing  with  girders  longitudinally  ;  beams 
crossways,  and  framed  into  or  resting  on  the 
girders ;  and  joists  framed  into  the  beams, 
running  the  same  direction  as  the  girders.  rig.  22. 

It  is  evident,  that  when  the  joists  are  not  flush  or  level  with  the  bottom 
of  the  beams  or  girders,  either  that  in  the  finish  the  beams  will  show, 
or  that  ceiling-joists  or  furrings  will  have  to  be  introduced. 

On  the  size  of  joists. — The  following  dimensions,  taken  in  part  from 
the  Liverpool  Building  Act,  may  be  considered  as  safe  sizes  for  ordinary 
constructions,  the  distances  from  centre  to  centre  being  one  foot. 

Joists  in  floors,  clear  bearing 

Exceeding  7  feet,  and  not  exceeding  10  feet,  to  be  not  less  than  6x2  inches. 

"  10     "  "  "  12     "  "  "          6  x  2J-  " 

,     "  12     "  «  "  14^  "  "  "         7  x  2J-  " 

"  14J-  "  "  "  16     "  "  "          8  x  2i  " 

"  1G     "  "  "  18     "  «  "          9  x  2|  " 

"  18     "  "  "  20     "  «  "  10  x  2£  " 

"  20     "  "  "  22     "  «  "  11  x  3  " 

»  22     "  «  "  24     "  "  "  12  x  3  " 

It  is  to  be  observed  that  lumber  is  seldom  sawed  to  dimensions  of  frac- 
tions of  an  inch  ;  we  must  therefore  adopt  a  width  of  an  integral  inch,  and 
proportion  the  distances  from  centre  to  centre,  according  to  the  increase  or 
decrease  of  width  given  to  the  joists. 

Trimmer  beams  and  headers  should  be  of 
greater  width  than  the  other  beams,  depend- 
ing on  the  distance  of  the  headers  from  the  wall, 
and  the  number  of  tail  beams  framed  into  it. 
The  New  York  Building  Act  requires  all  headers 
should  be  hung  in  stirrup  irons  (fig.  23),  and  not  framed  in.  It  also  re- 


Fig.  23. 


218 


ARCHITECTURAL   DRAWING. 


quires  all  girders  to  be  not  less  than  10  by  12  inches  square,  and  that  the 
posts  supporting  them,  shall  be  placed  at  intervals  of  not  more  than  10 
feet. 

Floors. — In  New  York  it  is  usual  to  lay  single  floors,  of  tongued  and 
grooved  boards,  but  in  the  Eastern  States,  double  floors  are  more  common ; 
the  first  floor  consists  of  an  inferior  quality  of  boards,  unmatched,  laid 
during  the  progress  of  the  work  as  a  sort  of  staging  for  the  carpenter  and 
mason,  and  in  finishing,  a  second  course  is  laid  on  them  of  better  material, 
generally  tongued  and  grooved,  but  ^sometimes  only  jointed.  Ceilings 
should  always  be  furred ;  that  is,  laths  should  never  be  nailed  directly 
to  the  joists;  the  usual  furrings  are  of  inch  board,  two  inches  wide, 
and  twelve  inches  from  centre  to  centre,  nailed  across  from  joist  to 
joist. 

Fig.  2-i.  represents  a  section  of  a  mill  floor.  The  girders  or  beams, 

generally  in  pairs  with  a  space 
of  about  an  inch  between  them, 


are  placed  at  a  distance  of  from 
Fig.  24.  seven  to  nine  feet  from  centre 

to  centre,  and  are  of  from  twelve  to  sixteen  inches  in  depth.  On  these,  a 
rough  plank  floor  of  from  three  to  four  inches  thick  is  laid ;  the  plank 
are  dowelled  together,  that  is,  put  together  with  pins  or  dowels,  like  a 
barrel  head.  Above  the  plank  is  laid  the  usual  top  floor,  and  beneath  a 
sheathing  of  thin  boards. 

For  extended  bearings  and  for  heavy  loads,  it  is  often  found  necessary 
to  truss  the  girders  or  beams.     Fig.  25  represents  a  bracing  truss  of 


Fig.  25. 


wrought  iron  between  a  double  girder ;  often  a  simple  piece  of  arched  iron 
is  let  into  the  wood,  half  on  each  side,  and  the  beams  bolted  strongly  to- 


Fig.  26. 


gether.    Fig.  26  represents  a  truss  by  suspension  ;  in  this  case,  the  strength 
depends  upon  the  cohesive  force  of  the  iron. 


AKCHITECTTJKAL    DRAWING.  219 

Fire-proof  floors. — Fig.  27  represents  a  section  of  the  fire-proof  flooring 


Fig.  27. 

constructed  by  Cooper  &  Hewitt.  The  girders  or  beams  are  of  wrought 
iron,  with  arches  of  a  single  course  of  brick  in  cement  between  them, 
resting  on  their  lower  flanches.  The  seven-inch  deep  beams  are  placed  at 
a  distance  of  from  three 'to  five  feet  from  centre  to  centre ;  extreme  width 
of  span,  between  side  walls,  fifteen  feet.  Strips  of  plank  are  fastened 
lengthways  at  the  side  or  on  top  of  the  beams,  to  receive  the  floor.  Fur- 
rings  for  the  ceiling  may  be  attached  crossways  to  the  bottom  of  the  beams, 
or  the  soffits  of  the  arches  may  be  plastered  without  any  preparation. 
Fig.  28  represents  a  section  of  one  of  the  French  systems  of  fire-proof 

floors.       It     Consists    Of     J 

girders,  placed  at  a  dis- 
tance of  one  metre  (39.38 
inches),  from  centre  to 
centre,  slightly  cambered  Fig.  28. 

or  curved  upwards  in  the  centre ;  the  depth  of  the  girders  to  depend 
upon  the  span.  Stirrups  of  cast  iron  are  slid  upon  the  girders,  into  which 
the  ends  of  flat  iron  joists,  set  edgeways,  pass  and  are  secured  by  pins ; 
the  ends  of  the  joists  take  a  bearing  also  on  the  bottom  flanges  of  the 
girders.  The  joists  are  placed  at  a  distance  of  one  metre  from  centre  to 
centre.  Upon  the  joists  rest  rods  of  square  iron,  which  in  this  way  form  a 
grillage  for  the  support  of  a  species  of  rough  cast  and  the  ceiling.  By 
this  and  other  very  similar  systems,  the  French  have  succeeded  in  reducing 
the  cost  of  such  floors  to  that  of  wooden  ones. 

Floors  are  sometimes  constructed  of  brick  in  single  or  in  groined 
arches,  the  thrust  being  opposed  by  the  weight  of  the  abutments,  but 
owing  to  its  expensiveness,  and  the  amount  of  room  occupied  by  the  ma- 
terial, this  kind  of  construction  is  not  at  present  very  common  in  edifices. 

Partitions  are  usually  simply  studs  set  at  intervals  of  twelve  or  six- 
teen inches,  these  spaces  being  adapted  to  the  length  of  the  lath  (forty- 
eight  inches).  The  sizes  of  the  studs  are  generally  2  x  4,  3  x  5,  or  3  x  6 
inches,  according  to  the  height  of  the  partition ;  for  any  high  partitions, 
greater  depth  may  be  required  for  the  studs,  but  three  inches  will  be  suf- 
ficient width.  Partitions  should  be  bridged  like  floors  with  herring-bone 
bridging. 


220 


ARCHITECTURAL    DRAWING. 


'I      g-LL 


JJ LL 


Fig.  29  represents  the  frame  of  the  side  of  a  wooden  house,  in  which 

A  A  are  the  posts,  B 
the  plate,  C  C  girts  or 
interties,  ~D  D  braces,  E 
sill,  F  window  posts  or 
studs,  G  G  studs. 

Usual  dimensions  of 
timber  for  frame  of  com- 
mon dwelling  houses  :  — 
sills  6x8,  posts  4x8, 
studs  2  x  4  or  3  x  4, 
girts  6  x  the  depth  of 
floor  joists,  plates  4  x 
6,  the  floor  joists  (J  fig. 
30),  are  notched  into  the 
girts  ;  more  frequently 
the  girts  are  omitted. 


Tig.  29. 


Fig.  ML 


Fig.  31. 


ID  n 


The  studs  are  of  the  same  length 
as  the  posts,  and  the  floor  joists  are 
supported  by  a  board,  a,  3  or  4  x  1 
let  into  the  studs  (fig.  31),  and  firmly 
nailed  ;  the  joists  are  also  nailed 
strongly  to  the  studs.  The  posts  and 
studs  are  tenoned  into  the  sills  and 
girts.  Fig.  32  represents  a  tenon, 
&  c,  in  side  and  end  elevation,  and 
mortice,  a;  the  portions  of  the  end 
of  the  stud  resting  on  the  beam 
are  called  the  shoulders  of  the 
tenon. 

Hoofs.  —  The  roofs  of  city  dwellings  and  stores  are  generally  flat,  that 
is,  with  but  very  little  inclination,  from  half  an  inch  to  two  inches  per  foot, 
merely  sufficient  to  discharge  the  water.  The  beams  are  laid  from  wall  to 
wall,  the  same  as  floor  timbers,  but  usually  of  less  depth,  or  at  greater  dis- 
tances between  centres,  and  with  one  or  two  rows  of  bridging.  The  roof 
is  laid  with  tongued  and  grooved  boards,  and  mostly  covered  with  tin. 

Figs.  1,  2,3,  PI.  XLV.  represent  side  or  portions  of  side  elevations  of 
the  usual  form  of  framed  roofs.  The  same  letters  refer  to  the  same  parts 
in  all  the  figures  of  the  plate.  T  T  are  the  tic,  beams,  E  R  the  main  rafters^ 


Fig.  32. 


AECHITECTCEAL   DRAWING.  221 

rr  the  jack  rafters,  P  P  the  plates,  pp  the  purlines,  KK  the  king  posts, 
k  k  king  bolts,  q  q  queen  bolts,  botli  are  called  suspension  bolts,  C  C  the 
collar  or  straining  beams,  B  B  braces  or  studs,  b  b  ridge  boards,  c  c  cor- 
bels. 

The  pitch  of  the  roof  is  the  inclination  of  the  rafters,  and  is  usually 
designated  in  reference  to  the  span  as  \,  1,  f ,  &c.,  pitch,  that  is,  the  height 
of  the  ridge  above  the  plate  is  £,  |,  |,  &c,  of  the  span  of  the  roof  at  the 
level  of  the  plate.  The  higher  the  pitch  of  the  roof,  the  less  the  thrust 
against  the  side  walls,  the  less  likely  the  snow  or  water  to  lodge,  and  con- 
sequently, the  tighter  the  roof.  For  roofs  covered  with  shingles  or  slate, 
in  this  portion  of  the  country,  it  is  not  advisable  to  use  less  than  i  pitch ; 
above  that,  the  pitch  should  be  adapted  to  the  style  of  architecture 
adopted.  The  pitch  in  most  common  use  is  1  the  span. 

Fig.  1  represents  the  simplest  framed  roof;  it  consists  of  rafters,  resting 
upon  a  plate  framed  into  the  ceiling  beam ;  this  beam  is  supported  by  a 
suspension-rod,  k,  from  the  ridge,  but  if  supported  from  below,  this  rod 
may  be  omitted.  This  form  of  construction  is  sufficient  for  any  roof  of 
less  than  25  feet  span,  and  of  the  usual  pitch,  and  may  be  used  for  a  40 
feet  span  by  increasing  the  depth  of  the  rafters  to  12  inches ;  deep  rafters 
should  always  be  bridged.  By  the  introduction  of  a  pmiine  extending 
beneath  the  centre  of  the  rafter,  supported  by  a  brace  to  the  foot  of  the 
suspension  rod,  as  shown  in  dotted  line,  the  depth  of  the  rafters  may  ob- 
viously be  reduced.  It  often  happens  that  the  king-bolt  may  interfere  with 
the  occupancy  of  the  attic ;  in  that  case  the  beam  is  otherwise  supported. 
Again,  it  may  be  necessary  that  the  tie  beam,  which  is  also  a  ceiling  and 
floor  beam,  should  be  below  the  plate  some  2  to  4  feet ;  in  that  case,  the 
thrust  of  the  roof  is  resisted  (fig.  4)  by  bolts,  l>  b,  passing  through  the 
plate  and  the  beam,  and  by  a  collar  plank,  C,  spiked  on  the  sides  of  the 
rafters,  high  enough  above  the  beam  to  afford  good  head  room.  For  roofs 
|  pitch  and  under  20  feet  span,  the  bolts  are  unnecessary,  the  collar  alone 
being  sufficient. 

Fig.  2  represents  a  roof,  a  larger  span  than  fig.  1  ;  the  frame  may  be 
made  very  strong  and  safe  for  roofs  of  60  feet  span.  King-bolts  or  sus- 
pension-rods are  now  oftener  used  than  posts,  with  a  small  triangular 
block  of  hard  wood  or  iron,  at  the  foot  of  the  bolts,  for  the  support  of  the 
braces.  The  objection  to  this  form  of  roof  is  that  the  framing  occupies 
all  the  space  in  the  attic ;  on  this  account  the  form,  fig.  3,  is  preferred  for 
roofs  of  the  same  span,  and  is  also  applicable  to  roofs  of  at  least  75  feet 
span,  by  the  addition  of  a  brace  to  the  rafter  from  the  foot  of  the  queen- 
bolt.  The  collar  beam  (fig.  6),  is  also  trussed  by  the  framing  similar  to 


222  ARCHITECTURAL   DRAWING. 

fig.  2.     In  the  older  roofs,  queen  posts  are  used  (fig.  33),  with  the  foot 
secured  by  straps  or  joint  bolts  to  the  tie  beam. 

In  many  church  and  barn  roofs  the  tie  beam  is  cut  off, 
fig.  5  ;  the  queen  post  being  supported  on  a  post,  or  itself 
extending  to  the  base,  with  a  short  tie  rod  framed  into  it 
from  the  plate. 

Figs.  7  and  8  are  representations  of  the  feet  of  rafters 
on  an  enlarged  scale.  In  fig.  7,  the  end  of  the  rafter  does  not 
project  beyond  the  face  of  the  plate  ;  the  coving  is  formed 
by  a  small  triangular,  or  any  desirable  form  of  plank, 
Fig.  33.  framed  into  the  plate.  The  form  given  to  the  foot  of  the 
rafter  is  called  a  crowfoot.  In  fig.  8,  the  rafter  itself  projects  beyond  the 
plate  to  form  the  coving.  Fig.  9  represents  a  front  and  side  elevation  and 
plan  of  the  foot  of  a  main  rafter,  showing  the  form  of  tenon,  in  this  case 
double  ;  a  bolt  passing  through  the  rafter  and  beam  retains  the  loot  of  the 
former  in  its  place.  Fig.  10  represents  the  side  elevation  of  the  foot  of  a 
main  rafter  with  only  a  small  portion  of  the  beam,  the  remainder  being 
supplied  by  a  rod.  In  fig.  7,  of  a  similar  construction  to  fig.  1,  the  tie  rod 
passes  directly  through  the  plate.  In  general,  when  neither  ceiling  nor 
flooring  is  supported  by  the  tie  beam,  a  rod  is  preferable. 

Roofs  are  now  very  neatly  and  strongly  framed  by  the  introduction  of 
cast-iron  shoes  and  abutting  plates  for  the  ends  of  the  braces  and  rafters. 
Fig.  11  represents  the  elevation  and  plan  of  a  cast-iron  king  head  for  a 
roof  similar  to  fig.  2.  Fig.  12,  that  of  the  brace  shoe  ;  fig.  13,  that  of  the 
rafter  shoe  for  the  same  roof.  Fig.  14,  the  front  and  side  elevation  of  the 
queen  head  of  roof  similar  to  fig.  3,  and  fig.  15, 
elevation  and  plan  of  queen  brace  shoe. 

Fig.  34  represents  the  section  of  a  rafter  shoe 
for  a  tie  rod ;  the  side  flanches  are  shown  in  dotted 
rig.  34.  line. 

On  the  size  and  the  proportions  of  the  different  members  of  a  roof: — 
Tie  beams  are  usually  intended  for  a  double  purpose,  and  are  therefore  af- 
fected by  two  strains ;  one  in  the  direction  of  their  length  from  the  thrust 
of  the  rafters,  the  other  a  cross  strain,  from  the  weight  of  the  floor  and 
ceiling.  In  estimating  the  size  necessary  for  the  beam  the  thrust  need  not 
be  considered,  because  it  is  always  abundantly  strong  to  resist  this  strain, 
and  the  dimensions  are  to  be  determined  as  for  a  floor  beam  merely, 
each  point  of  suspension  being  a  support.  When  tie  rods  are  used,  the 
strain  is  in  the  direction  of  their  length  only,  and  their  dimensions  can  be 
calculated,  knowing  the  pitch,  span,  and  weight  of  the  roof  per  square 


ARCHITECTURAL   DRAWING.  223 

foot  and  the  distance  apart  of  the  ties,  or  the  amount  of  surface  retained 
by  each  tie. 

Rule. Multiply  one  half  the  weight  of  the  roof  by  one  half  the  span, 

and  divide  the  product  by  the  pitch. 

Example. What  is  the  strain  upon  the  tie  rod  of  a  roof  40  feet  span 

and  15  feet  pitch  ? 

The  weight  of  the  wood-work  of  the  roof  may  be  estimated  at  40  Ibs. 
per  cubic  foot,  or  on  an  average  at  about  12  Ibs.  per  foot  square,  slate  at  7 
to  9  Ibs.,  shingles  at  1£  to  2  Ibs.  The  force  of  the  wind  may  be  assumed 
at  15  Ibs.  per  square  foot.  The  excess  of  strength  in  the  timbers  of  the 
roof  as  allowed  in  all  calculations,  will  be  sufficient  for  any  accidental  and 
transient  force  beyond  this.  If,  therefore,  the  roof  be  like  fig.  1,  Plate 
XLV.,  without  ceiling  beneath,  and  retained  by  a  tie  rod,  we  may  consider 
as  the  weight  per  square  foot  for  a  slate  roof:  12  +  7  +  15  =  34  Ibs.  The 
length  of  the  rafter  is  ^20*  +  15r=:  25  feet;  hence,  if  the  tie  rods  are  10 
feet  apart,  the  amount  of  surface  on  each  incline,  or  one  half  of  the  roof, 
supported  by  the  tie  is  25  x  10  =  250  square  feet,  which  multiplied  by 
the  weight  per  square  foot,  or  250  x  34  =  8,500  Ibs. ;  applying  the  rule, 
8500  x  20  -5- 15  =  11,333  Ibs.,  the  thrust  on  the  tie  rod.  If  we  estimate  the 
strength  of  wrought-iron  at  10,000  Ibs.  per  square  inch  of  section,  or  8,000 

1 1 33S 

Ibs.  when  a  thread  is  cut  upon  the  end,  then,  —    —  =  1.416  square  inches, 

8000 

or  a  rod  a  little  exceeding  1T5F  inches  in  diameter. 

The  rafters  (fig.  1,  PI.  XLV.)  may  be  considered  as  jack  rafters  of  long 
bearings,  or  as  a  beam  supporting  transversely  the  weight  of  the  roof,  and 
the  accidental  pressures,  and  may  be  estimated  by  resolving  the  direction 
of  these  pressures  into  a  line  perpendicular  to  the  direction  of  the  rafters. 

The  pressure  on  main  rafters  (figs.  2  and  3)  is  in  the  direction  of  their 
length,  when  they  are  supported  by  braces  at  or  very  near  the  points  where 
the  purlines  rest ;  but  in  addition  to  the  weight  of  the  roof,  they  support  a 
portion  of  the  weight  of  the  tie  beam,  and  whatever  may  be  dependent 
upon  it.  If  the  frame  is  like  fig.  2,  that  is,  with  a  king^-bolt  or  post,  and 
the  weight  is  uniformly  distributed  upon  the  beam,  then  one  half  the 
weight  is  supported  by  the  bolt  or  post,  and  consequently  by  the  rafter, 
and  the  other  half  by  the  side  walls.  Under  the  same  circumstances,  the 
suspension  rods  (fig.  3),  support  each  £  of  the  weight  of  the  beam,  &c.,  and 
the  side  walls  each  £.  But  in  general,  where  the  attic  is  made  use  of,  the 
load  is  not  uniformly  distributed,  by  far  the  greatest  part  is  suspended 
upon  the  rods. 


224:  AKCHITECTUEAL   DK  AWING. 

To  find  the  pressure  on  the  main  rafter.  Multiply  one  half  the  weight 
of  the  roof,  and  that  portion  of  the  weight  of  the  beam  and  its  load  which 
may  depend  upon  it,  by  the  length  of  the  rafter,  and  divide  the  product 
by  the  pitch. 

Example. — What  is  the  pressure  upon  the  main  rafter  of  a  slate  roof 
of  56  feet  span,  21  feet  pitch,  frames  10  feet  between  centres,  and  form 
like  fig.  3,  with  an  uniformly  distributed  load  on  the  beam  of  8,400  Ibs., 
and  a  load  between  the  suspension  rods  of  10,000  Ibs.  ? 


The  length  of  rafter  is     */28'  +  21'  =  35  feet. 

Assuming  the  load  per  square   foot  upon  the  rafter  at  40  Ibs.  per 
square  foot, 

then  40  x  31  x  10  =  12,400  Ibs.,  £  the  weight  of  roof. 

i  of  the  uniform  weight  — =    2,800  Ibs. 

1 0000 
\  the  weight  between  rods  *  —  =    5,000  Ibs. 


Total,  20,200  Ibs. 
20200_x35=33)6(,61te_ 

If  we  now  assume  the  resistance  of  wood  at  740  Ibs.  per  square  inch,*  or 
600  Ibs.  for  length  exceeding  13  times  their  thickness, 

—  =  56.111  square  inches  of  section. 
oUU 

The  proportion  of  the  depth  to  the  width  is  generally  about  10  to  8  or  5 
to  4. 

Hence  |/|?  x  5  =  1.67  x  5  =  8.35  inches  =  depth. 

1.67  x  4  =  6.68  inches  —  width. 

Gwilt,  in  his  Architecture,  recommends  the  following  dimensions  for 
portions  of  a  roof: 

*  Weisbach. 


ARCHITECTURAL   DRAWING. 


225 


SPAN. 

FORM   OF   EOOF. 

RAFTERS. 

BRACES. 

POSTS. 

COLL'R  BEAMS. 

feet. 

inches. 

inches. 

inches. 

inches. 

25 

Fig.  2,  Plate  XLVI. 

5x4 

5x3 

5x5 

30 

u                « 

6x4 

6x3 

6x6 

35 

Fig.  3,        " 

5x4 

4x2 

4x4 

7x4 

45 

" 

6x5 

5x3 

6x6 

7x6 

50 

2  Sets  of  Queen  Posts 

8xG 

5x3 

(    8x8) 

9x6 

(8x4) 

GO 

14                                     « 

8x8 

6x3 

j  10  x  8  ) 

11  x  6 

(  10  x  4  1 

These  dimensions,  for  rafters,  are  somewhat  less  than  the  usual  practice 
in  this  country  ;  no  calculations  seem  to  have  been  made  for  using  the  attic. 
An  average  of  common  roofs  here,  would  give  the  following  dimensions 
nearly:  30  feet  span,  8x5  inches;  40  feet,  9  x  6 ;  50  feet,  10  x  7;  60 
feet,  11x8;  collar  beams  the  same  size  as  main  rafters.  Eoof  frames 
from  8  to  12  feet  from  centre  to  centre. 

Dimensions  for  jack  rafters  15  to  18  inches  apart. 


For  a  bearing  of  6  feet, 

<t  if  g          U 

"  "        10     « 

«  «         12     " 

«  «         20     " 


3  x  2|  inches. 
4x3        " 
5x3        « 
6x3        " 
10  x  3        « 


Purlines  : 


LENGTH    OF   BEARING. 

DISTANCES   APART   IN    FEET. 

feet. 

6 

8 

10 

12 

6i 

6    x    3£ 

61  x   4 

7x5 

8x5 

8 

7x5 

8x5 

9x5 

9x6 

10 

9x5 

10   x   5 

10   x    6 

11    x    6 

12 

10   x    6 

11    x   6 

12    x   7 

13   x    8 

The  pressure  on  the  plates  is  transverse  from  the  thrust  of  the  rafters, 
but  in  all  forms  except  fig.  1,  owing  to  the  notching  of  the  rafters  on  the 
purlines,  this  pressure  is  inconsiderable.  The  usual  size  of  plates  for  figs. 
1  and  2,  is  6  x  6  inches.  For  forms,  fig.  1,  the  size  depends  on  the  mag- 
nitude of  roof,  and  the  distance  between,  the  ties.  The  width  in  all  such 
cases  to  be  greater  than  the  depth ;  4  to  6  inches  may  be  taken  as  the 
depth,  8  to  12  for  the  width. 

Joints. — As  timber  cannot  always  be  obtained  of  sufficient  length  for 

the  different  portions  of  a  frame,  it  is  often  necessary  to  unite  two  or  more 
15 


226 


ARCHITECTURAL   DRAWING. 


pieces  together  by  the  ends,  called  scarfing  or  lapping.     Figs.  35  and  36 
are  the  most  common  means  of  lapping  or  halving, 
(          J~r~l         /       which  methods  may  be  employed  when  there  is 
not  much  longitudinal    compression   or  extension. 
When  such  an  effect  is  to  be   provided  for,  the 
upper  as  well  as  the  lower  timber   should  be  let 
Fig.  86.  into  each  other.     Figs.  37,  38,  39,  40,  41,  are  differ- 


Fig.  85. 


Fig.  40.  Fig.  41. 

ent  methods  to  obtain  this  result.  In  figs.  37,  38,  the  joint  is  brought  to  a 
bearing  by  a  key  driven  in  tight.  Fig.  39  represents  a  scarf  suited  for  a 
beam  supported  at  this  joint  by  a  post,  and  where  there  is  tensile  strain, 
the  timber  should  be  joint-bolted  or  anchored.  The  centre  of  the  post 
should  be  beneath  the  extreme  edge  of  the  lower  joint.  Figs.  40  and  41 
are  long  scarfs,  in  which  the  parts  are  bolted  through  and  strapped,  suited 
for  tie  beams.  Joints  are  also  often  made  by  abutting  the  pieces  together 
and  bolting  splicing  pieces  on  each  side ;  still  further  security  is  given  by 
cutting  grooves  in  both  timbers  and  pieces  and  driving  in  keys. 

Varieties  of  Roofs. — Roofs  of  country  edifices  especially  are  considera- 
bly varied  in  outline  ;  without  central  supports  the  forms  of  frames  are  but 
modified  examples  of  those  already  given  ;  with  central  supports,  many,  like 
the  roofs  of  city  buildings,  may  be  considered  as  the  framing  of  floors,  or  the 
sidings  of  wooden  houses.  City  roofs  are  generally  composed  of  beams  laid 
from  wall  to  wall,  or  girder  to  girder,  with  a  slight  incline  of  from  %"  to  1" 
per  foot,  with  the  drip  and  gutter  at  the  rear,  with  the  front  finished  with 
a  cornice,  and  sometimes  a  false  French  roof,  or  a  show  of  a  steep  pitch. 

Square  houses  are  often  framed  with  hipped  roofs,  that  is,  with  a  pitch 
on  each  side,  as  at  A  and  B,  figs.  42  and  43,  which  are  called  hips,  whilst 
C  and  D  are  gables.  The  fig.  43  represents  the  plan  of  a  roof  as  usually 
drawn  in  strong  or  close  line  or  deep  shade  at  the  ridge,  and  lighter  at  the 


ARCHITECTURAL    DRAWING. 


227 


eaves.  The  Gambrel  or  Mansard  roof  (fig.  44)  is  a  roof  with  two  kinds  of 
pitch  ;  the  theory  of  its  construction  is  that  of  the  polygon  of  rods  (p.  118), 
but  in  general  they  have  central  support  from  partitions,  and  their  outlines 
are  much  varied  by  curves  in  the  lower  rafters  cut  from  plank. 


Fig.  48. 


\ 


The  roofs  of  cotton  and  woollen  factories  are  often  framed  with  two 
pitches,  with  a  small  upright  part  at  the  angle  for  windows,  extending 
the  whole  length  of  the  roof.  It  may  be  framed  as  two  roofs,  the  lower, 
as  in  fig.  3,  Plate  XLY,  and  the  upper,  plain  rafters  projecting  suffi- 
ciently beyond  the  purline,  and  over  the  roof,  to  receive  the  window- 
frame. 

In  the  framing  of  roofs  it  has  been  the  author's  practice  of  late  years  to 
disregard  entirely  purlines,  jack -rafters,  and  plates,  and  make  the  covering 
of  plank  2"  to  3",  according  to  space  between  frames,  tongued  and  grooved, 
and  spiked  strongly  to  the  main  rafters.  The.  roofs  of  the  gate-houses  at 
New  Croton  Reservoir  are  modified  examples  of  this ;  with  plates  trussed 
laterally,  as  the  angles  of  the  roofs  were  cut  by  ventilating  towers. 

Circular  Roofs. — Circular  roofs  have  often  been  constructed  of  strips 
of  boards,  cut  to  the  width  of  the  rafters,  bent  into  the  form  of  the  projected 
arc,  and  nailed  to  the  depth  required  for  the  span  of  the  roof.  In  northern 
climates  they  are  objectionable  on  account  of  the  flatness  of  the  top  and  the 
unequal  distribution  of  a  load  of  snow.  Very  large-span  cylindrical  roofs 
have  been  made  for  station-houses  of  Howe's  truss  (fig.58),  with  circular 
chords.  Small  cylindrical  roofs,  say  not  exceeding  30  feet  span,  are  otten 
made  of  curved  and  corrugated  iron,  with  tie-bolts  just  above  the  eaves. 

Eaves  of  roof  are  finished  with  cornices,  of  various  mouldings  appro- 
priate to  the  style  of  the  building.  The  gutters  of  eave-troughs  are  gene- 


228 


ARCHITECTURAL   DRAWING. 


rally  formed  in  the  cornice  (fig.  45)  ;  sometimes  on  the  top  of  the  roof  (fig. 
46),  and  sometimes  by  raising  a  parapet  (fig.  47),  and  forming  a  valley. 


Fis.  !<i.  Fig.  45.  Fig.  47. 

Iron  Roofs. — PL  XL VI.  fig.  1,  represents  the  half  elevation  of  an  iron 
roof  of  a  forge  at  Paris ;  figs.  2,  3,  4,  details  on  a  larger  scale.  This  is  a 
common  form  of  iron  roof,  consisting  of  main  rafters,  E,  of  the  J  section, 
fig.  4,  trussed  by  a  suspension  rod,  and  tied  by  another  rod.  The  pmiines 
are  also  of  I  iron,  secured  to  the  rafters  by  pieces  of  angle  iron  on  each 
side ;  and  the  roof  is  covered  with  either  plate  iron  resting  on  jack  rafters, 
or  corrugated  iron  extending  from  purlin e  to  puiiine.  The  rafter  shoe,  A, 
and  the  strut,  S,  are  of  cast  iron,  all  the  other  portions  of  the  roof  are  of 
wrought  iron. 

The  surface  covered  by  this  particular  roof,  is  53  metres  (164  feet)  long, 
and  30  metres  (98£  feet)  wide.  There  are  11  frames,  including  the  two  at 
the  ends,  which  form  the  gables. 

The  following  are  the  details  of  the  dimensions  and  weights  of  the  dif- 
ferent parts : 

Ibs. 

2  rafters,  0.72  feet  deep,  length  together,  99.1  feet,                 .             .             .  1,751 

5  rods,  0.13  feet  diameter,      "         "        131.4    "...  882 

16  holts,             " 79 

8  bridle-straps,  0.24  x  .05             ......  123 

2  pieces,  .046  thick,  connecting  the  rafters  at  the  ridge, •> 

4       "  "  at  the  foot  of  the  strut   .  .       >' 

4       "      .036  thick,  uniting  the  rafters  at  the  junction  in  the  strut — together 

with  their  bolts  and  nuts,              .             .             .             .              .  1 76 

2  cast  iron  struts,               .......  308 

2  rafter  shoes,               .                                       .....  287 

Total  of  one  frame,  .  .  .  .  .  .  .  3,695 

16  purlines,  1  ridge  iron,  each  0.46  deep,  17.2  long,     ....        2,985 

Bolts  for  the  same,  .......  64 

16  jack  rafters,  I  iron,  0.16  deep,        ......        2,489 

Weight  of  iron  covering,  including  laps,  per  square  foot,       .  .  2.88 


ARCHITECTURAL   DRAWING.  229 

The  weight  of  iron  in  this  roof  could  be  reduced  by  substituting  corru- 
gated iron  for  the  covering,  even  of  less  weight  per  square  foot,  and  omit- 
ting the  jack  rafters. 

Roofs  are  sometimes  made  with  deep  corrugated  main  rafters  with  flat 
iron  between,  or  purlines  and  corrugated  iron  for  the  covering.  The  great 
objection  to  iron  roofs,  lies  in  the  condensation  of  the  interior  air  by  the 
outer  cold,  or,  as  it  is  termed,  sweating ;  on  this  account  they  are  seldom 
used  for  other  buildings  than  boiler  houses  or  depots,  except  a  ceiling  be 
made  below  to  prevent  the  contact  of  the  air  inside  with  the  iron. 

Fig.  5  is  an  elevation  of  nearly  two  of  the  three  panels  of  one  of  the 
cast  iron  girders  for  connecting  the  columns,  and  carrying  the  transverse 
main  gutters,  which  supported  the  roof  of  the  English  Crystal  Palace. 
Figs.  6,  7,  8,  9,  10,  11,  sections  of  various  parts  on  an  enlarged  scale. 

The  depth  of  the  girder  was  3  feet,  and  its  length  was  23  ft.  3f  inches. 
The  sectional  area  of  the  bottom  rail  and  flange  in  the  centre  (fig.  7),  was 
6J-  square  inches  ;  the  width  of  both  bottom  and  top  rail  (fig.  6),  was  re- 
duced to  3  inches  at  their  extremities.  It  will  be  observed  that  the  section 
of  the  braces  and  ties  are  such  as  to  give  great  stiifness,  and  the-  section  of 
the  braces  at  ~b  ~b  (fig.  8),  is  greater  than  at  c  c  (fig.  9) ;  the  section  of  the 
tie  (fig.  10),  is  the  same  as  the  brace  at  c  c ;  they  are  all  formed  with  a 
draft,  that  is,  with  a  taper  from  the  centre  to  the  outside  of  from  ^  to  TV 
of  an  inch  on  a  side,  according  to  the  depth  of  the  feather. 

The  weight  of  these  girders  was  about  1,000  Ibs.,  and  they  were  proved 
by  a  pressure  of  9  tons,  distributed  on  the  centre  panel. 

A  second  series  of  girders  were  made  of  similar  form  to  fig.  5,  but  of 
increased  dimensions  in  the  section  of  their  parts.  Their  weight  averaged 
about  1,350  Ibs.,  and  they  were  proved,  as  above,  to  15  tons. 

A  third  series,  of  increased  section  of  parts,  weighed  about  2,000  Ibs.,* 
and  were  proved  to  22£  tons. 

Fig.  12  represents  an  elevation  of  two  of  the  nine  panels  of  one  of  the 
wrought  iron  trusses  which  carry  the  lead  flat  and  arched  roof  across  the 
nave  of  the  Crystal  Palace.  These  trusses  are  72  feet  long  and  6  feet  deep. 
The  top  rail  G,  shown  in  section  fig.  13,  consists  of  two  angle  irons  4^ 
inches  deep,  3^  inches  wide,  and  |  of  an  inch  thick,  with  a  plate  9  inches 
wide  and  f  thick,  riveted  on  top.  A  space  of  2  inches  is  left  between  the 
angle  irons.  The  angle  irons  are  in  five  lengths,  and  are  connected  by 
eight  £  rivets  passing  through  them,  and  through  a  plate  or  plates  intro- 
duced between  them.  The  top  plate  is  in  seven  lengths,  connected  by  i 
inch  rivets,  through  the  angle  irons,  the  plate,  and  a  joint  plate.  The  top 
plate  is  riveted  to  the  angle  irons  by  1  inch  countersunk  rivets,  5  inches 


230  ARCHITECTURAL    DRAWING. 

apart.  The  bottom  rail  consists  of  two  flat  wrought  iron  bars,  6  inches 
deep,  with  a  2  inch  space  between  them.  It  is  in  four  lengths,  jointed  by 
six  1  inch  rivets  passing  through  joint  plates  6  x  18  x  jf  inches  on  the 
outside,  and  three  plates  17  X  f  inches.  The  bars  forming  the  central 
lengths  of  the  bottom  rail  are  £  of  an  inch  thick,  and  those  forming  the 
side  lengths  are  f  of  an  inch  thick. 

The  end  standards  are  of  cast  iron,  3£  inches  wide,  4  inches  deep,  and 
1  inch  thick,  of  a  T  form  of  section,  secured  to  the  column  by  six  l£  inch 
bolts.  The  standard  is  2  inches  thick  at  top  and  bottom  where  it  receives 
the  rails.  Two  sockets  are  formed  in  the  middle,  to  receive  the  diagonals 
I  and  J.  I,  being  exposed  to  compression,  is  made  of  four  angle  irons, 
2i  x  2i  x  TV  inch,  riveted  together  in  pairs  with  ±  inch  rivets.  The 
diagonal  J  is  formed  of  two  bars,  4£  x  \  inch,  and  is  secured  at  each  end 
by  a  1  f  inch  rivet.  The  ends  are  thickened  by  short  plates  riveted  to  them 
to  make  up  in  a  measure  the  loss  of  strength  from  the  large  rivet  hole. 
The  diagonal  K  is  formed  of  two  bars  4|  inches  deep  by  1  inch  thick,  and 
is  fixed  at  each  end  by  a  2  inch  bolt  and  nut.  The  other  diagonals,  being 
exposed  to  much  less  strain,  are  formed  of  single  bars  4|  x  \  inch,  and 
are  secured  at  each  end  by  a  1  inch  rivet. 

The  standards  B  and  C  consist  each  of  four  angle  irons,  2i  x  2i  x  \ 
inch,  riveted  together  in  pairs,  and  the  two  pairs  riveted  together  with  six 
small  cast  iron  distance  pieces  between  them.  The  next  standard,  that  is, 
the  third  from  each  end,  but  not  shown  in  the  drawing,  is  of  cast  iron.  It 
is  of  +  section,  being  at  the  centre  6x6  inches,  thickness  of  metal  f  to  £ 
of  an  inch.  The  base,  which  rests  upon  the  base  of  the  bottom  rail,  is  18 
x  4  inches,  and  the  top  is  18  x  3  inches.  Triangular  projections  enter 
the  top  and  bottom  rail,  where  they  are  secured  by  1  inch  rivets.  In  the 
Centre  is  a  socket  or  slot  through  which  pass  the  two  light  diagonals. 
The  main  strength  of  the  truss  consists  in  the  top  and  bottom  rails,  the 
diagonals  I,  J,  K,  the  first  wrought  iron  standard,  B,  and  the  cast  iron 
standard,  D. 

On  the  General  Principles  of  Bracing. — Let  fig.  42  be  the  elevation 
of  a  common  roof  truss,  and  let 
a  weight,  W,  be  placed  at  the 
foot  of  one  of  the  suspension 
rods.  Now,  if  the  construction 
consisted  merely  of  the  rafter 
C'  B,  and  the  collar-beam  C'  C,  F1g- 43- 

resting  against  some  fixed  point,  then  the  point  B  would  support  the  whole 
downward  pressure  of  the  weight ;  but  in  consequence  of  the  connection 


ARCHITECTURAL    DRAWING. 


231 


of  the  parts  of  the  frame,  the  pressure  must  'be  resolved  into  components 
in  the  direction  C'  A  and  C'  B,  C'  5  will  represent  the  pressure  in  the 
direction  C'  B,  C'  w  the  portion  of  the  weight  supported  at  B,  C'  a  the 
pressure  in  the  direction  C'  A,  and  w  W  the  portion  of  the  weight  sup- 
ported on  A.  The  same  resolution  obtains  to  determine  the  direction  and 
amount  of  force  exerted  on  a  bridge  truss  of  any  number  of  panels,  by  a 
weight  placed  at  any  point  of  its  length  (fig.  43.)  In  either  case,  the 
effect  of  the  oblique  form  C'  A,  upon  the  angle  C,  is  evidently  to  force 
r.  c'  it  upwards ;  that  is,  a  weight  placed  at  one 

side  of  the  frame  has,  as  in  case  of  the  arch, 
a  tendency  to  raise  the  other  side.    The  effect 
Fis-  ^  of  this  upward  force  is  a  tension  on  a  por- 

tion of  the  braces,  according  to  the  position  of  the  weight ;  but  as  braces, 
from  the  manner  in  which  they  are  usually  connected  with  the  frame,  are 
not  capable  of  opposing  any  force  of  extension,  it  follows  that  the  only 
resistance  is  that  which  is  due  to  the  weight  of  a  part  of  the  structure. 

Figs.  44  and  45  illustrate  the  results  of  overloading  at  single  points 
such  forms  of  construction. 


Fig.  45. 


Fig.  46. 


To  remedy  this  effect,  if  counter  braces  be  introduced,  as  shown  in  dot- 
ted lines  (fig.  46),  the  tendency  of  a  weight  moving  across  the  structure  is 
to  compress  the  counters  and  extend  the  braces.  But  since,  as  we  have 
said,  braces  are  not  usually  framed,  especially  in  wooden  structures,  to  re- 
sist a  tensile  strain,  it  is  necessary  to  overcome  this  force  in  another  way ; 
that  is,  by  introducing  wedges  between  the  end  of  the  counter  braces  and 
the  joints  against  which  they  abut,  or  by  means  of  the  counters  and  the 
suspension  rods,  in  any  way  straining  the  structure  so  that  there  may  be 
an  additional  compression  upon  the  brace  more  than  the  upward  or  tensile 
force  exerted  by  any  passing  weight.  In  this  case,  therefore,  the  passage 
of  a  load  would  produce  no  additional  strain  upon  any  of  the  timbers,  but 


232  ARCHITECTURAL   DRAWING. 

would  tend  to  relieve  the  counters.  The  counter  braces  do  not,  of  course, 
assist  in  sustaining  the  weight  of  the  structure;  on  the  contrary,  the 
greater  the  weight  of  the  structure  itself,  the  more  will  the  counter  braces 
be  relieved. 

If,  instead  of  the  counter  braces,  the  braces  themselves  are  made  to  act 
both  as  tie  and  as  a  strut,  as  has  been  done  sometimes  in  iron  bridges  and 
trusses,  then  the  upward  force  will  be  counteracted  by  the  tension  of  the 
brace,  but  counter  braces  are  preferable,  as  it  is  better  that  the  force 
exerted  against  any  portion  of  the  structure  should  always  be  in  one  direc- 
tion. 

It  follows,  from  what  has  been  shown  of  the  eifect  of  a  variable  load, 
that  no  bridge,  either  straight  or  arched,  intended  for  the  passage  of  heavy 
vehicles  or  trains,  should  ever  be  without  counter  braces  or  diagonal  ties. 

On  the  Truss  ~by  Tension  Rod  (fig.  53). — Since  the  limit  of  the  elas- 

a, ._*.. ,«•'..  I 


Fig.  53. 

ticity  of  iron  is  very  small  in  comparison  with  wood,  when  iron  is  thus 
used  to  truss  timber,  the  rods  must  break  before  the  beam  reaches  the  de- 
flection that  the  weight  should  produce.  It  is  evident,  therefore,  that  in 
construction  the  beam  should  not  be  cambered  by  the  tension  of  the 
rod,  but  that  the  top  of  the  beam  should  be  arched,  and  be  permitted  to 
settle  with  the  weight  before  it  strains  the  rod  at  all.  The  rods  should  be 
depended  on  to  resist  the  whole  of  the  tension.  To  estimate  the  strain 
upon  the  "suspension  rod,  multiply  the  weight  supported  at  the  point  c  or  of 
by  the  length  of  the  rod  a  d  or  d'  5,  and  divide  the  product  by  the  length 
of  the  strut  c  d.  The  length  of  the  rod  and  of  the  strut  may  be  measured 
from  any  horizontal  line  which  completes  the  triangle. 

Suppose  a  system  to  be  composed  of  a  series  of  suspension  trusses,  as 
in  fig.  54,  in  which  the  load  is  uniformly  distributed.  If  we  represent  the 
load  at  each  of  the  points,  4,  3,  2,  1,  2'  &c.,  by  1,  the  load  at  4  will  be 
supported  3  upon  a  and  |  upon  3 ;  hence  the  strut  3  will  have  to  support 
a  load  of  1  +  .5  =  1.5 ;  of  this,  f  will  be  supported  by  2  and  %  by  a ;  f 
of  1.5=1,  1  +  1=2,  load  on  strut  2 ;  £  of  this  load,  or  1.5,  will  be  supported 
at  1,  and  since  from  the  opposite  side  there  is  an  equal  force  exerted  at  1, 
therefore  the  strut  1  supports  1  +  1.5  +  1.5  =  4;  the  tension  on  the  rod 
c  2  is  4;  on  2  3,  4  +  1  =  5;  on  3  4-,  5  +  1  =  6;  on  4  a,  6  +  1  =  7;  and 
the  rod  should  therefore  be  increased  in  strength  in  these  proportions  from 


ARCHITECTURAL    DRAWING. 


233 


the  central  point  c,  to  the  point  of  suspension,  a.     The  tension  on  the  rods 


3  4,  2  3,  1  2,  may  be  easily  resolved  from  their  direction  and  the  load  upon 
the  several  struts. 

If  this  construction  be  reversed,  the  parts  which  now  act  as  ties  must  be 
made  as  braces,  and  braces,  ties ;  then  we  have  a  roof  truss,  and  the  force 
exerted  on  the  several  parts  may  be  estimated  in  a  similar  way  as  for  the 
suspension  truss. 

To  frame  a  construction  so  that  it  may  be  completely  braced,  that  is, 
under  the  action  of  any  arrangement  of  forces,  the  angles  must  not  admit 
of  alteration,  and  consequently  the  shape  cannot.  The  form  should  be 
resolvable  into  either  of  the  following  elements : — Figs.  55,  56,  57. 


Fig.  55. 

In  these  figures,  lines 
compression ;    lines     - 


Fig.  56. 


Fig.  57. 


represent  parts  required  to  resist 
parts    to    resist    tension    only ;    lines 
parts  to  resist  both  tension  and  compression. 
In  a  triangle  (fig.  55),  an  angle  cannot  increase  or  diminish,  without 
the  opposite  angles  also  diminishing  or  increasing.     In  the  form  fig.  56,  a 
diagonal  must  diminish ;  in  fig.  55  a  diagonal  must  extend,  in  order  that 
any  change  of  form  may  take  place.     Consequently  all  these  forms  are 


\ 


Fig.  58. 


Fig.  59. 


completely  braced,  as  each  does  not  permit  of  an  effect  taking  place,  which 
would  necessarily  result  from  a  change  of  figure.     Hence  also,  any  system 


234:  ARCHITECTURAL   DRAWING. 

composed  of  these  forms,  properly  connected,  breaking  joint  as  it  were 
into  each  other,  must  be  braced  to  resist  the  action  of  forces  in  any  direc- 
tion ;  but  as  in  general  all  bridge  trusses  are  formed  merely  to  resist  a 
downward  pressure,  the  action  on  the  top  chord  being  always  compression, 
it  is  not  necessary  that  these  chords  should  act  in  both  capacities.  As 
illustrations  of  bridge  trussing,  a  few  panels,  fig.  58,  of  Howe's  truss,  and 
of  Pratt's,  fig.  59,  are  given. 

On  the,  Size  and  Proportion  of  Rooms  in  general. — "  Proportion  and 
ornament,"  according  to  Ferguson,  "  are  the  two  most  important  resources 
at  the  command  of  the  architect,  the  former  enabling  him  to  construct  or- 
namentally, the  latter  to  ornament  his  construction."  A  proportion  to  be 
good,  must  be  modified  by  every  varying  exigence  of  a  design,  it  is  of 
course  impossible  to  lay  down  any  general  rules  which  shall  hold  good  in 
all  cases  ;  but  a  few  of  its  principles  are  obvious  enough.  To  take  first 
the  simplest  form  of  the  proposition,  let  us  suppose  a  room  built,  which 
shall  be  an  exact  cube — of  say  20  feet  each  way — such  a  proportion  must 
be  bad  and  inartistic ;  (and  besides,)  the  height  is  too  great  for  the  other 
dimensions.  As  a  general  rule,  a  square  in  plan  is  least  pleasing.  It  is 
always  better  that  one  side  should  be  longer  than  the  other,  so  as  to  give  a 
little  variety  to  the  design.  Once  and  a  half  the  width  has  been  often 
recommended,  and  with  every  increase  of  length  an  increase  of  height  is 
not  only  allowable,  but  indispensable.  Some  such  rule  as  the  following 
meets  most  cases  :  "  The  height  of  the  room  ought  to  be  equal  to  half  its 
width  plus  the  square  root  of  its  length ; "  but  if  the  height  exceed  the 
width  the  effect  is  to  make  the  room  look  narrow;  again,  by  increasing  the 
length  we  diminish,  apparently,  the  other  two  dimensions.  This,  however, 
is  merely  speaking  of  plain  rooms  with  plain  walls ;  it  is  evident  that  it  will 
be  impossible,  in  any  house,  to  construct  all  the  rooms  and  passages  to 
conform  to  any  one  rule  of  proportion,  nor  is  it  necessary,  for  in  many 
rooms  it  would  not  add  to  their  convenience,  which  is  often  the  most  desir- 
able end ;  and  if  required,  the  unpleasing  dimensions  may  be  counteracted 
by  the  art  of  the  architect,  for  it  is  easy  to  increase  the  apparent  height  by 
strongly  marked  vertical  lines,  or  bring  it  down  by  horizontal  ones.  Thus 
if  the  walls  of  two  rooms  of  the  same  dimensions  be  covered  with  the  same 
strongly  marked  striped  paper,  in  one  case  the  stripes  being  vertical,  and 
in  the  other  horizontal,  the  apparent  dimensions  will  be  altered  very  con- 
siderably. So  also  a  deep  bold  cornice  diminishes  the  apparent  height  of  a 
room.  If  the  room  is  too  long  for  its  other  dimensions,  this  can  be  reme- 
died by  breaks  in  the  walls,  by  the  introduction  of  pilasters,  &c.  So  also, 
as  to  the  external  dimensions  of  a  wall,  if  the  length  is  too  great  it  is  to  be 


AECHITECTUEAL    DRAWING.  235 

remedied  by  projections,  or  by  breaking  up  the  lengths  into  divisions. 
This  will  be  understood  by  reference  to  elevations  of  "  Country  Houses," 
Plates  LXV.  to  LX1X.  In  this  view,  as  variety  in  form  adds  greatly  to  the 
picturesque,  it  is  far  better  in  designing  a  country  house,  where  one  is  not 
restricted  to  room,  to  mark  out  the  rooms  to  the  size  which  we  wish  them  to 
be,  cutting  out  slips  of  paper  of  the  dimensions,  according  to  some  scale,  to 
arrange  them  then  in  as  convenient  an  order  as  possible,  and  again  modifying 
the  arrangement  by  the  necessities  of  construction  and  economy.  Thus 
the  more  the  enclosing  surface,  in  proportion  to  the  -included  area,  the 
greater  the  number  of  chimneys,  unnecessary  extent  of  passages;  all  of 
course  conduce  to  an  excess  of  expense.  Again,  the  kitchen  should  be  of 
convenient  access  to  the  dining  room,  both  should  have  large  and  commo- 
dious pantries,  and  all  rooms  should  have  an  access  from  a  passage,  with- 
out being  compelled  to  pass  through  another  room ;  this  is  particularly 
applicable  to  the  communication  of  the  kitchen  with  the  front  door.  Out- 
side doors  for  common  and  indiscriminate  access  should  open  into  passages 
and  never  into  important  rooms. 

As  to  the  size  of  the  different  rooms,  they  must  of  course  depend  on 
the  purposes  to  which  they  are  to  be  applied,  the  class  of  house,  and  the 
number  of  occupants.  To  commence  with  the  kitchen,  for  the  poorer  class 
of  houses  it  is  also  used  as  an  eating  room,  and  should  therefore  be  of  con- 
siderable size  to  answer  both  purposes ;  for  the  richer  houses,  size  is  neces- 
sary for  the  convenience  of  the  work ;  in  New  York  City  houses  the  aver- 
age will  be  found  to  be  about  15  x  18  feet,  for  medium  houses  in  the 
country  they  are  in  general  less,  say  12  x  14.  A  back  kitchen,  scullery, 
or  laundry,  should  be  attached  to  the  kitchen,  arid  may  serve  as  a  passage 
way  to  out  of  doors. 

The  dining  or  eating  rooms. — The  width  of  dining  tables  vary  from  3 
to  5  feet  6  inches,  the  depth  occupied  by  the  chair  and  person  sitting  at  the 
table  is  about  18  inches  on  each  side  ;  the  table  space,  for  comfort,  should 
be  not  less  than  2  feet  for  each  person  at  the  sides  of  the  table,  and 
considerable  more  at  the  head  and  foot ;  hence  we  may  calculate  the  space 
that  will  be  necessary  for  the  family  and  its  visitors,  at  the  table.  If  we 
now  allow  a  farther  space  of  2  feet  at  each  side  for  passages,  and  some  3 
to  5  at  the  head  for  the  extra  tables  or  chairs,  we  can  mark  out  the  mini- 
mum of  space  required :  but,  if  possible,  do  not  confine  the  dining  room  to 
meagre  limits,  unless  for  very  small  families ;  let  not  the  parties  be  lost  in 
the  extent  of  space,  nor  let  them  appear  crowded. 

The  show  room  parlors,  if  there  are  any  intended  for  such  in  the  house, 
may  be  made  according  to  the  rules  given  above,  not  square,  but  the  length 


236 


ARCHITECTURAL    DRAWING. 


about  once  and  a  half  the  width ;  if  much  longer  than  this,  break  up  the 
walls  by  transoms  or  projections.  As  to  the  particular  dimensions  no  rules 
can  be  given,  it  must  depend  on  every  person's  taste  and  means.  20  x  16 
may  be  considered  a  fair  medium  size  for  a  regular  living  room  parlor,  not 
a  drawing  room.  The  same  size  will  answer  very  well  for  a  sleeping  room. 
The  usual  width  of  single  beds  is  2  feet  8  inches,  of  three-quarter  3  feet 
6  inches,  of  whole  4  feet  6  inches,  the  length  6  feet  6  inches,  so  that  if 
adequate  means  of  ventilation  are  provided,  it  is  easy  to  see  into  how  small 
quarters  persons  may  be  thrust.  The  bed  should  not  stand  too  near  the 
fire,  nor  between  two  windows ;  its  most  convenient  position  is  head  against 
an  interior  wall,  with  a  space  on  each  side  of  at  least  2  feet. 

Pantries. — Closets  for  crockery  should  not  be  less  than  14  inches  in 
width  in  the  clear ;  for  glass  8  inches,  and  for  the  hanging  ^up  of 
clothes,  not  less  than  18  inches.  For  medium  houses,  the  closets 
of  large  sleeping  rooms  should  be  at  least  3  feet  wide,  with  hanging  room, 
and  drawers  and  shelves.  There  should  also  be  blanket  closets,  for  the 
storing  of  blankets  and  linen ;  these  should  be  accessible  from  the  entries, 
and  may  be  in  the  attic.  Store  closets  should  also  be  arranged  for  gro- 
ceries and  sweetmeats. 

Passages. — The  front  entries  are  usually  6  feet  wide  in  the  clear ;  com- 
mon passage  ways  are  usually  3  feet  wide ;  these  are  what  are  required,  but 
ample  passages  give  an  important  effect  to  the  appearance  of  the  houses. 
The  width  of  principal  stairs  should  be  not  less  than  3  feet,  and  all  first 
class  houses,  especially  those  not  provided  with  water-closets  and  slop 
sinks  on  the  chamber  floor,  should  have  two  pair  of  stairs,  a  front  and 
a  back  pair ;  the  back  stairs  may  not  necessarily  be  over  2  feet  6  inches 
in  width. 

The  Height  of  Stories. — It  is  usual  to  make  the  height  of  all  the  rooms 
on  each  floor  equal,  it  can  be  avoided  by  furring  down,  or  by  the  break- 
ing up  of  the  stories,  by  the  introduction  of  a  mezzonine  or  intermediate 
story  over  the  smaller  rooms.  Both  remedies  are  objectionable  ;  the  more 
artistic  way  is  to  obviate  the  appearance  of  disproportionate  height  by 
means  stated  above. 

The  average  height  of  the  stories  for  such  city  dwellings  as  we  have 
given  plans  of  are :  cellar  6  feet  6  inches,  common  basement  8  to  9  feet, 
English  basement  9  to  10,  principal  story  12  to  15,  first  chamber  floor  10 
to  12,  other  chamber  floors  8  to  10  feet,  all  in  the  clear.  For  country 
houses  the  smaller  of  the  dimensions  are  more  commonly  used.  Attic 
stories  are  sometimes  but  a  trifle  over  6  feet  in  height,  but  are  of  course 
objectionable. 


AKCHITECTURALte  DRAWING. 


237 


Fig.  60. 


Details  of  parts.  Stairs  consist  of  the  tread  or  step  on  which  we  set 
our  feet,  and  risers,  upright  pieces  supporting  the  treads — each  tread  and 
riser  forms  a  stair.  If  the  treads  are  parallel  they  are  called  fliers,  if  less 
at  one  end  than  the  other,  they  are  called  winders;  f  and  w,  fig.  62.  The 
top  step,  or  any  intermediate  wide  step,  for  the  purpose  of  resting,  is  called 
a  landing.  The  height  from  the  top  of  the  nearest  step  to  the  ceiling 
fa  above  is  called  the  headway.  The  rounded  edge  of 
the  step  is  called  a  nosing,  a,  tig.  60  ;  if  a  small  hol- 
low, 5,  be  glued  in  the  angle  of  the  nosing  and  riser, 
it  is  called  a  moulding  nosing.  The  pieces  which  sup- 
port the  ends  of  the  stairs  are  called  strings,  that 
against  the  wall  the  wall  string,  the  other  the  outer 
string,  ^Besides  the  strings,  pieces  of  timber  are  framed  and  placed  beneath 
the  fliers,  called  carriages.  The  opening  on  plan  (which  must  occur  be- 
tween the  outer  strings,  if  they  are  not  perpendicular  over  each  other)  is 
called  the  well  hole,  W,  fig.  62. 

The  breadth  of  stairs  in  general  use  is  from  9  to  12  inches.  In  the  best 
staircases,  the  breadth  should  never  be  less  than  12  inches,  nor  more  than 
15.  The  height  of  the  riser  shoul  d 
be  the  more,  the  less  the  width  of 
the  tread ;  for  a  15  inch  tread  the 
riser  should  be  5  inches  high,  for 
12  inches,  6f,  for  9  inches,  8.  In 
laying  out  the  plan  of  stairs,  hav- 
ing determined  the  starting  point 
either  at  bottom  or  top  as  the  case 
may  be,  find  exactly  the  height  of 
the  story ;  divide  this  by  the  height 
you  suppose  the  riser  should  be. 
Thus  (fig.  61),  if  the  height  of 
the  story  and  thickness  of  floor  be 
9  feet,  and  we  suppose  the  riser 
should  be  7  inches  high,  then 
108  inches,  divided  by  7  =  15f . 
It  is  clear  that  there  must  be 
an  even  number  of  steps,  either  16  or  15 ;  to  be  near  to  what  we  have  sup- 
posed to  be  the  height  of  the  riser,  adopt  15,  then 

-y-8-  =  7T3j  inches,  height  of  riser. 

For  this  particular  case  we  assume  the  breadth  of  the  step  as  10  inches, 
and  the  length  at  3  feet,  a  very  usual  length,  seldom  exceeding  4  feet  in 


238 


AECI1ITECTU  RAL    DK  A  WING. 


the  best  staircases  of  private  bouses.  For  tbe  plan — lay  off  tbe  outside  of 
the  stairs,  two  parallel  lines  3  feet  apart,  and  space  off  from  the  point  of 
beginning  14  treads  of  10  inches  each,  and  draw  the  cross  parallel  lines. 

To  construct  the  elevation,  the  line  of  the  stair  in  plan  mav  be  pro- 
jected, and  the  height  be  divided  into  the  number  of  risers,  15  of  7}  inches 
each,  and  cross  parallels  drawn  through  these  points ;  or  the  same  points 
may  be  determined  by  intersection  of  the  projections  of  the  plan  with  a 
single  inclined  line  drawn  through  the  nosing  of  top  and  bottom  steps. 
It  is  to  be  observed  that  the  number  of  treads  is  always  one  less  than  the 
number  of  risers,  the  reason  of  which  will  appear  by  observing  the  elevation. 
The  drawing  of  the  elevation  of  stairs  is  in  general  necessary,  to  deter- 
mine the  opening  to  be  framed  in  the  upper  floor,  to  secure  proper  head- 
way. Thus  (fig.  61),  the  distance  between  the  nearest  stair  and  the  ceiling 
at  a  should  not  be  less  than  6  feet  6  inches ;  a  more  J||||fL_ 
ample  space  improves  the  look  of  the  stairway;  '  \S— i 
but  if  we  are  confined  in  our  limits,  this  will  de- 
termine the  position  of  one  trimmer,  the  other  will 
be  of  course  at  the  -top  of  the  stairs.  "When  one 
flight  is  placed  over  another,  the  space  required  for 
timber  and  plastering,  under  the  steps,  is  about  6 
inches  for  ordinary  stairs. 

When  the  stairs  are  circular,  or  consist  in  part 
of  winders  and  fliers,  as  in  fig.  62,  the  width  of  the 
tread  of  the  winders  should  be  measured  on  the 
central  line.  The  construction  of  the  elevation  is 
similar  to  that  of  the  straight  run  (fig.  61),  by  divid- 
ing the  space  between  the  stories  by  a  number  of 
parallel  lines  equal  to  the  number  of  risers,  and  in- 
tersecting the  parallels  by  projections  from  the  plan. 
Fig.  63  represents  a  circular 
flight  of  stairs  without  a  well 

hole,  the  narrow  ends  of  the  winders  being  mortised  into 
a  central  shaft  or  newel,  N.  The  same  term  is  also  ap- 
plied to  the  first  laluster  or  post  of  the  hand  rail.  The 
objection  to  all  circular  stairs  of  this  form,  or  with  a 
small  well  hole,  is  that  there  is  too  much  difference  be- 
tween the  width  of  the  tread,  but  a  small  portion  being  of  a  suitable  size. 
The  handsomest  and  easiest  stairs  are  straight  runs,  divided  into  landings, 
intermediate  of  the  stories,  and  either  continuing  then  in  the  same  line,  or 
turning  at  right  angles,  or  making  a  full  return. 


Fig.  62. 


ARCHITECTURAL   DRAWING. 


239 


The  top  of  the  hand  rail  should,  in  general,  be  about  2  feet  8  inches 
above  the  nosing,  and  should  follow  the  general  line  of  the  steps.  The 
angles  of  the  head  rail  should  always  be  eased  off.  A  hand  rail,  affording 
assistance  in  ascending  or  descending,  should  not  be 
wider  than  the  grasp  of  the  hand,  fig.  64 ;  but  where, 
for  architectural  effect,  a  more  massive  form  may  be 
necessary,  it  is  very  convenient,  and  may 
be  very  ornamental,  to  have  a  sort  of 
double  form,  that  is,  a  smaller  one 
planted  on  top  of  the  larger,  fig.  65.  Fig.  64.  Kg.es. 

J)oors. — Fig.  60  represents  the  elevation,  and  fig.  67  the  horizontal  sec- 
tion of  a  common  inside  door.  A  A  are  the  stiles,  B,  C,  II,  D,  the  bottom, 
lock,  parting,  and  top  rail,  E  the  panels,  and  F  the  muntmj  the  combina- 
tion of  mouldings  and  offsets  around  the  door,  G,  is  called  the  architrave / 
in  the  section,  a  a  are  the  partition  studs,  5  b  the  door  jambs. 

With  regard  to  the  proportions  of  internal  doors,  they  should  de- 
pend in  some  degree  on  the  size  of  the  apartments;  in  a  small  room 
a  large  door  always  gives  it  a  diminutive  appearance,  but  doors  lead- 
ing from  the  same  entry,  which  are  brought  into  the  same  view,  should 

be  of  uniform  height.  The  smaller 
doors  which  are  found  on  sale  are  2  ft. 
4  in.  x  6  feet;  for  water  closets,  or 
very  small  pantries,  they  are  some- 
times made  as  narrow  as  20  inches, 
but  any  less  height  than  6  feet  will 
not  afford  requisite  head  room.  2  ft. 
9  in.  x  7  ft.,  3  ft.  x  7  ft.  6  in.,  or  3  ft. 
6  in.  x  8  ft.,  are  well  proportioned  6 
panelled  doors.  But  the  apparent 
proportions  of  a  door  may  be  varied 
by  the  omission  of  the  parting  rail, 
making  the  door  4  panelled,  or  nar- 
rowed still  more  by  the  omission  of 
the  lock  rail,  making  a  2  panelled 
door.  Sometimes  the  muntin  is  omit- 
ted, making  but  one  panel ;  but  this  of 
course  will  not  add  to  the  appearance 
of  width,  but  the  reverse.  Wide  panels 
i  are  objectionable,  as  they  are  apt  to 

Fig.  67.  "   shrink  from  the  mouldings  and  crack. 


Fig.  66. 


240 


AKCHITECTDKAL    DRAWING. 


When  the  width  of  the  door  exceeds  5  feet,  it  is  generally  made  in  two 
parts,  each  part  being  hung  to  its  side  of  the  frame,  or  one  part  hung  to 
the  other,  so  as  to  fold  back  like  a  shutter ;  or  the  parts  may  be  made  to 
slide  back  into  pockets  or  grooves  in  the  partition,  as  shown  in  plan  and 
horizontal  section,  figs.  68  and  69.  One  of  the  doors  in  the  drawing  is 
shown  as  a  sash  door,  the  other  close  panels,  so  as  to  give  twp  illustrations 
in  the  same  diagram  ;  the  same  may  be  said  of  the  architrave.  It  may  be 
unnecessary  to  say  that  in  construction  both  sides  and  doors  should  be  uni- 
form. The  upper  panels  of  the  close  door  may  be  made  of  glass  ;  the  finish 
around  this  half  of  the  door  is  with  an  architrave,  as  in  fig.  66,  but  with 


Fig.  69. 


different  mouldings.  The  finish  over  the  other  half  of  the  door  is  an  entab- 
lature, supported  by  pilasters  A,  commonly  called  by  carpenters  antse,  though 
not  correctly  so,  the  antse  being  pilasters  at  the  end  of  a  projecting  wall. 

Figs.  70  and  71  are  the  elevation  and  horizontal  section  of  an  antse- 
finished  outside  door,  with  the  side  lights  C  C,  and  a  top,  fan,  or  transom 
light  B.  The  bar  A  is  called  a  transom,  and  this  term  is  applied  generally 
to  horizontal  bars  extending  across  openings,  or  even  across  rooms. 

Fig.  72  is  the  elevation  of  an  outside  folding  door.  The  plan  (fig.  73) 
shows  a  vestibule  ~V,  and  an  interior  door.  The  outer  doors,  when  open, 
fold  back  into  the  pockets  or  recesses,  p  p,  in  the  wall.  This  is  the  present 


ARCHITECTURAL    DRAWING, 


241 


usual  form  of  doors  for  first-class  houses  in  this  city.     The  fan  lights  are 

made  semicircular,  and  also  the  head  of  the  upper  panels  of  the  door ; 

these  panels  in  the  interior  or  vestibule  door  are  of  glass. 

Windows  are  apertures  for  the  admission  of  light  to  the  building,  for 

ventilation,  and  for  looking  out.  When  used  for  the  admission  of  light 
only,  the  sashes  may  be  stationary,  as  they 
sometimes  are  in  churches,  but  for  most  po- 
sitions they  are  intended  for  all  these  pur- 
poses, and  therefore  the  sashes  are  made  to 
open,  either  by  sliding  vertically,  or  laterally, 
or  like  doors.  The  first  is  the  common  form 
of  window,  the  sashes  are  generally  balanced 
by  weights;  the  second,  except  in  a  cheap  form 
in  mechanics'  shops,  are  seldom  used ;  the  third 
are  called  casements,  or  French  windows. 

Figs.  74 
and  75  repre- 
sent in  eleva- 
tion and  plan 
the  parts  of 
the  common 
sash  window 
and  its  shut- 
ters, in  bro- 
-ken  lines,  so 
as  to  show 
the  details  on 


Fig.  78. 

a  large  scale.     S  designates  the  sill  of  the 

sash  frame,  W  the  stone  sill,  with  a  wash 

to  discharge  the  water,  B  is  the  bottom  rail 

of  the  sash,  M  the,  meeting  rails,  and  T  the 

top  rail,  H  is  the  head  of  the  sash  frame, 

A  the  architrave  similar  to  that   around 

doors.     In  the  sectional  plan  C  C'  are  the 

window  stiles,  F  the  pulley  stile,  w  w  the 

sash  weights,  p  the  parting  strip,  and  D  D 

double  fold  shutters.      Sash  windows  for 

dwellings  are  almost  always  made  with  Fis-  75. 

twelve  lights,  six  in  each  sash.     The  height  of  the  window  must  of  course 

depend  on  the  height  of  the  room.     Unless  the  windows  begin  from,  or 
16 


242 


ARCHITECTURAL   DRAWING. 


nearly  from,  the  floor,  the  point  a  (fig.  74)  may  be  fixed  at  a  height  of 
about  30  inches  above  the  floor,  and  the  top  of  the  window  sufficiently  be- 
low the  ceiling  to  allow  space  for  the  architrave  or  other  finish  above  the 
window,  and  for  the  cornice  of  the  room,  if  there  be  any  ;  a  little  space  be- 
tween these  adds  to  the  effect.  For  common  windows,  the  width  of  the 
sash  is  4  inches  more  than  that  of  the  glass,  and  the  height  6  inches  more ; 
thus  the  sash  of  a  window  3  lights  wide  and  4  lights  high,  of  12  x  16  glass, 
is  3  feet  4  inches  wide,  and  5  feet  10  inches  high.  In  plate  glass  windows 
more  width  is  taken  for  the  stiles  and  rails.  The  most  usual  sizes  of  glass 
are  7  x  9,  8  x  10,  9  x  12,  10  x  12,  10  x  14,  11  x  15,  12  x  16,  12  x  18,  12  x  20, 
14  x  20,  but  glass  may  be  had  of  intermediate  or  of  much  larger  sizes.  Plate 
glass,  either  polished  or  rough,  may  be  had  of  a  size  as  large  as  14  x  7  feet. 
Fig.  76  represents  the  elevation  of  half  of  a  French  window,  also  in  broken 
lines,  the  same  letters  designate  similar  parts  as  in  fig.  68.  A  transom  bar 
is  often  framed  between  the  meeting  rails,  and  in  this  case  the  upper  sash 

may  be  movable  ;  in  the  fig.  it  is  fixed.   An  upright,  called 

a  mullwn,  is  often  introduced  in  the  centre,  against  which 

the  sash  shuts.     Fig.  77  is  an  elevation  on  a  smaller  scale. 
For  convenience  of  ^~ —  j 

egress  and  ingress,  the 

lower    sashes    should 

not  be  less  than  5  feet 

6  inches  high,  that  is, 

when  the  window  opens 

on  a  stoop  or  balcony. 

It  will  be  seen  that  in 

both  forms  of  sash  the 

bottom  rail  is  the  widest, 

and  that  for  the  same 

aperture     the     French 

window  admits  the  least 

light.     The  chief  objec- 
tion to  this  window  lies 

in  the  difficulty  of  keep- 
ing out  the  rain  at  the 

bottom  in  a  driving 
storm.  To  obviate  this,  the  small  mould- 
ing d,  with  a  drip  or  undercut,  is  nailed 
to  the  bottom  rail ;  but  the  more  effec- 
tual means  is  the  patent  weather  strip,  the  same  as  used  on  outside  doors. 


Fig.  76. 


Pig.  7T. 


ARCHITECTURAL   DRAWING. 


243 


The  most  simple  exterior  finish  for  windows,  in  brick  or  stone  houses, 
is  a  plain  stone  cap  and  sill,  the  height  of  the  cap  for  common  apertures 
being  from  four  to  five  courses  of  brick,  and  the  sill  three  courses,  the 
latter  always  to  project  from  one  to  two  inches  beyond  the  line  of 
brickwork.  Usually  in  wooden  structures,  and  often  in  stone  and  brick, 
an  architrave  is  formed  around  the  window  (figs.  78,  and  79).  For  brick 
houses  the  facings  are  made  of  stone.  The  architrave  should  not  project 
so  much  as  to  interfere  with  the  shutting  back  of  the  blinds.  Blinds  are 
commonly  tlutee-eighths  of  an  inch  narrower,  and  one  inch  longer  than 
the  sash. 


Fig.  ?a 


U  LI 

Fig.  79. 


Fig.  SO. 


Fig.  80  represents  a  section  of  the  finish  around  the  bottom  of  the  wall 
of  a  room.  A,  is  the  base,  consisting  of  a  plain  strip  or  skirting,  with  a 
moulding  above  it.  B,  is  the  surbase  or  chair  rail ;  between  these,  it  is 
not  unusual  to  have  a  panelled  or  plain  board,  called  the  dado.  The 
rough  plastering  is  usually  continued  to  the  floor,  the  skirting  and  surbase 
are  then  nailed  on,  the  hard  finish  is  next  put  on,  and  lastly  the  base 
moulding.  The  panels  of  the  dado  are  imitated  in  oil  or  distemper ;  the 
surbase  is  seldom  used  but  in  dining  rooms  or  offices. 

For  the  finish  of  the  angle  of  the  wall  and  ceiling,  the  cornice  adds 
often  to  the  architectural  effect.  It  consists  of  a  series  of  mouldings  similar 
to  the  style  of  finish  of  the  houses,  extending  around  the  room.  The 
effect  of  the  cornice  is  to  diminish  the  apparent  height  of  the  room ; 
for  low  rooms,  if  adopted,  it  should  extend  in  width  on  the  ceiling,  and 


244 


AKCHITECTTJKAL   DBAWING. 


but  little  in  depth  on  the  wall,  and  the  reverse  where  an  opposite  effect 
is  desired.     Figs.  81,  82,  83. 


Fig.  81. 


Fig.  82. 


Fig.  83. 


Fire-Places. — Fire-places  for  wood  are  made  with  flaring  jambs  of  the 
form  shown  in  plan  fig.  84 ;  the  depth  from  1  foot  to  15  inches,  the  width 
of  opening  in  front  from  2  feet  6  inches  to  4  feet,  according  to  the  size  of 
the  room  to  be  warmed ;  height  2  feet  3  inches  to  2  feet  9  inches,  the 
width  of  back  about  8  inches  less  than  in  front ;  but  at  present  fire-places 
for  wood  are  seldom  used,  stoves  and  grates  having  superseded  the  fire- 
place. The  space  requisite  for  the  largest  grate  need  not  exceed  2  feet  in 
width  by  8  inches  in  depth.  The  requisite  depth  is  given  by  the  projec- 


C 


Fig.  85. 


Fig.  86. 


tion  of  the  grate,  and  the  mantel-piece.  Ranges  require  from  4  feet  4  inches 
to  6  feet  4  inches  wide  x  12  inches  to  20  inches  deep  ;  jambs  8  inches  to 
12  inches.  Fig.  86  represents  the  elevation  of  a  mantel-piece  of  very  usual 
proportions.  The  length  of  the  mantel  is  5  feet  5  inches,  the  width  at 
base  4  feet  6  inches,  the  height  of  opening  2  feet  7  inches,  and  width  2  feet 
9  inches.  A  portion  of  this  opening  is  covered  by  the  iron  sides  or  archi- 
trave of  the  grate,  and  the  actual  open  space  would  not  probably  exceed 
18  inches  in  width  by  2  feet  in  height.  The  sizes  of  flues  are  8  x  8,  4  x  12, 
and  8  x  12  inches.  In  brick  or  stone  houses  the  flues  are  formed  in  the 
thickness  of  the  wall,  but  when  distinct  they  have  an  outside  shell  of  a 
half  brick  or  4  inches.  The  flues  of  different  fire-places  should  be  distinct, 
those  from  the  lower  stories  pass  up  through  the  jambs  of  the  upper  fire- 


ARCHITECTURAL   DRAWING.  24:5 

places,  and  keeping  side  by  side  with  but  4-inch,  brick  work  between  them, 
are  topped  out  above  the  roof,  sometimes  in  a  double  and  often  in  a  single 
line  16  inches  wide  by  a  breadth  required  by  the  number  of  flues.  The 
chimney  is  usually  capped  with  stone,  sometimes  with  tile  or  cement  pots. 
As  an  architectural  feature,  the  chimney  is  often  made  to  add  considerably 
to  the  effect  of  a  design. 

Privies,  Water-Closets,  and  Out-Houses. — The  size  of  privies  must 
depend  greatly  on  the  uses  of  the  building  to  which  they  are  to  be  at- 
tached, its  position,  and  the  character  of  its  occupants.  Allowing  nothing 
for  evaporation  or  absorption,  the  entire  space  necessary  for  the  excremen- 
titious  deposits  of  each  individual,  on  an  average,  will  be  about  seven 
cubic  feet  for  six  months,  of  which  three-quarters  is  fluid.  In  the  country, 
vaults  are  usually  constructed  of  dry  rubble-stone ;  and  the  fluid  matters 
are  expected  to  be  filtered  through  the  earth,  the  same  as  in  cess-pool 
waste ;  but  great  care  must  be  taken  that  they  neither  vitiate  the  water  supply 
nor  the  air  of  the  house.  A  brick  and  cement  vault,  air  and  water  tight, 
with  a  ventilating  pipe  into  a  hot  chimney-flue,  is  the  best  preventive, 
and  may  even  be  built  within  the  house.  In  all  other  cases  there  should 
be  free  air  space  between  the  house  and  privy.  In  the  city,  where  there 
is  adequate  water  supply  and  sewerage,  the  water-closet  should  be  adopted, 
except  in  houses  occupied  by  many  ignorant  and  irresponsible  tenants, 
who  throw  extraneous  matters  into  the  hoppers,  and  obstruct  the  sewer- 
pipes.  In  these,  tight  privy-vaults,  with  trapped  sewer  connections,  and 
with  all  the  house  waste  and  roof  water  discharging  into  them,  are  the 
easiest  kept  in  order.  Water-closet,  or  privy,  with  a  single  seat,  should 
occupy  a  space  not  less  than  4  ft.  x  2  ft.  6  in.  The  rise  of  seat  should 
be  about  17  in.  high ;  and  the  hole  egg-shaped,  11  x  8  in. 

Cess-Pools. — The  house  waste,  when  there  ia  no  system  of  sewerage, 
can  only  be  got  rid  of  by  cess-pools,  which  permit  its  absorption  in  the 
earth,  or  by  cistern,  and  using  the  water  for  irrigation  and  manuring. 
Their  size  will  depend  entirely  on  the  quantity  of  this  waste.  They 
should  be  placed  as  far  as  possible  from  the  house,  and  the  connec- 
ctions  with  them  should  be  trapped,  that  no  effluvia  may  escape  into  the 
house. 

Traps  are  of  various  kinds.  The  general  principle  of  their  construc- 
tion is  by  bend  or  basin  in  the  sewer-pipe  to  make  a  water  closure,  through 
which  water  may  pass,  but  not  air. 

For  Wood  or  Coal  Sheds. — In  estimating  the  size  of  these  accessories, 
it  may  only  be  necessary  to  state  that  a  cord  of  wood  contains  128  cub.  ft., 
and  a  ton  of  coal  occupies  a  space  of  about  40  cub.  ft. 


24:6  ARCHITECTURAL   DRAWING. 


DRAWING. 

The  reader  is  probably  now  sufficiently  acquainted  with  the  use  of 
drawing  instruments  to  construct,  with  but  little  explanation,  most  archi- 
tectural drawings.  The  first  thing  to  be  done  toward  the  execution  of  a 
drawing  is  to  determine  the  scale  upon  which  the  drawing  is  to  be  made. 
The  usual  scale  for  plans  and  elevations  in  architectural  drawings  is 
either  4  ft.  or  8  ft.  to  the  inch,  and  especially  in  working  drawings  it 
is  necessary  that  the  scale  should  be  on  some  marked  division  of  the  com- 
mon two-foot  rule.  Working  details  are  generally  drawn  on  coarse  paper, 
and  on  as  large  a  scale  as  possible,  often  full  size. 

To  construct  the  Ist-story  plan  (fig.  87),  end  (fig.  88),  and  side  (fig.  89) 
elevation  of  a  house.  Select  a  scale  of  say  4  ft.  to  the  inch,  and  commence 
with  the  plan.  Lay  off  a  base  line  A  B,  on  this  measure  of  20  ft. 
for  the  length  of  the  house,  and  erect  perpendiculars  at  the  extrem- 
ities thus  measured.  Lay  off  16  ft.  for  the  width  on  each  of  the  perpen- 
diculars, and  connect  these  points ;  the  line  will  be  parallel  to  A  B,  and  the 
outline  of  the  house  will  be  defined.  The  thickness  of  the  wall  for  a  house 
of  this  size,  if  of  brick,  will  be  8  inches  for  wall  and  2  inches  for  furring 
and  plastering,  or  10  inches ;  if  of  wood,  the  studs  should  be  2  x  4 ;  \  \ 
inches  for  outside  boarding  and  1  for  lath  and  plaster,  or  6^  in  total.  Lay 
off  on  the  inside  of  the  outline  the  thickness  of  the  wall,  and  draw  the  in- 
terior lines.  Lay  off  the  partitions  marking  the  rooms,  by  a  single  line 
or  by  two  lines.  The  thickness  of  partition  for  such  an  edifice  will  be 
from  4  to  5  inches.  The  dimensions,  as  generally  marked,  should  be  from 
outside  lines  to  centres  of  partitions,  or  from  centres  to  centres  of  parti- 
tions, as  more  determinate  for  the  carpenter  to  work  from  than  dimensions 
in  the  clear,  that  is,  actual  spaces  in  the  rooms.  Lay  off  the  position  of  the 
windows,  which  are  to  be  3  ft.  wide,  preserving  uniformity  both  in  inside 
and  outside  appearance.  Thus,  the  front  window  should  be,  as  near  as 
possible,  in  the  centre  of  the  room,  and  on  the  outside,  uniform  with  the 
door.  The  side  of  the  door  may  be  4  ft.  from  the  end  of  the  house  ;  mark 
its  position  and  width ;  but  the  side  of  the  window,  if  in  the  centre  of  the 
room,  will  be  5  ft.  4  in.  from  the  opposite  corner ;  make  it  5  ft.,  and  lay 
off  the  window  in  the  rear  opposite  to  the  front  window.  The  openings 
for  windows  are  distinguished  from  doors  by  straight  lines  drawn  across 
the  aperture.  The  window  in  the  end  of  the  house,  to  present  a  uniform 
appearance  outside,  should  be  in  the  centre,  and  this  suits  the  purpose  for 
which  the  room  is  designed,  a  blank  corner  being  necessary  for  the  bed. 


AECHITECTUEAL    DKAWING. 


247 


Lay  off  the  position  of  the  fireplace  in  the  centre  of  the  end  of  the 
room,  the  opening  to  be  3  ft,  the  width  of  the  jambs  8  in.  each,  and 
the  width  of  the  back  2  ft.  4  in.  Draw  lines  for  a  partition  flush  with 
the  face  of  the  chimney  to  one  side  of  the  room,  with  an  opening  for 
a  door  to  be  for  a  closet  or  pantry ;  all  the  inside  doors  to  be  2  ft.  10  in. 
wide,  and  to  be  represented  by  openings  in  the  partitions  merely.  Lay 


off  the  stairs,  as  shown  in  the  lobby,  2  ft.  9  in.  wide.  Fill  the  space  be- 
tween the  interior  and  exterior  outlines  of  the  walls  and  partition  in  black, 
leaving  the  openings  for  the  doors  and  windows,  and  the  plan  is  complete. 
The  filling  is  not  indispensable,  but  it  adds  to  the  distinctness  of  the  plan. 


248  AECHITECTUKAL   DRAWING. 

To  construct  the  front  elevation  (fig.  88),  project  the  various  points  or 
position  of  the  corners,  window  and  door,  as  shown  on  the  plan,  and  extend 
the  lines  of  projection  as  high  as  may  be  necessary  above  the  line  CD; 
on  these  lines  mark  off  the  height  of  window  and  door,  the  height  of  the 
eaves  in  projecting  must  be  determined  from  the  end  elevation  ;  finish  the 
window  and  door  with  lines  to  represent  caps  and  sills,  and  whatever  other 
lines  may  be  necessary  for  the  style  of  finish  of  the  drawing.  On  the  line 
C  D  erect  also  the  outlines  of  the  end  elevation,  taking  the  horizontal  di- 
mensions from  the  plan,  and  projecting,  as  far  as  possible,  the  vertical  ones 
from  the  front  elevation.  Establish  the  ridge  of  the  roof  by  setting  off 
the  height  of  the  pitch  on  the  centre  line  above  the  eaves,  and  draw  the 
various  lines  to  represent  the  boarding  and  cornice.  Draw  the  chimney 
in  the  centre  above  the  ridge.  It  is  to  be  observed  that  this  chimney  is  at 
the  opposite  end  of  the  house,  and  may  be  represented  in  lighter  lines  than 
.  the  rest  of  the  drawing.  The  base  of  the  chimney  and  the  eaves  are  to 
be  projected  on  the  front  elevation.  It  is  to  be  observed  that  many  lines 
may  be  projected,  either  from  the  front  to  the  side  elevation,  or  vice  versa ; 
the  true  way  is  to  construct  both  elevations  together. 

The  outline  of  the  section  may  be  taken  from  the  end  elevation,  and 
the  other  plans  from  Ist-story  plan,  or  constructed  directly  from  the  dimen- 
sions. 

Plates  XL VII.  to  LI.  inclusive  are  plans  and  elevations  of  a  house,  and 
contain  as  full  representations  as  are  usually  given  by  architects  for  the 
purposes  of  estimate,  accompanied  by  specifications.  The  size  of  our  page 
has  compelled  the  titles  to  be  put  within  the  body  of  the  drawings  ;  place 
them  outside,  and  give  good  margin.  On  Plate  LI.  the  section  and  end 
elevation  are  given  together.  This  is  also  for  economy  of  space;  but 
should  be  copied  by  the  draughtsman  in  two  distinct  drawings. 

Page  249  represents  plans  of  familiar  forms  of  houses,  all  drawn  to  the 
same  scale,  as  illustrations  to  the  student,  and  as  examples  to  be  copied  on 
a  larger  scale.  The  same  letters  of  reference  are  used  on  all  the  plans,  for 
rooms  intended  for  similar  purposes.  Thus,  K  K  designate  kitchens,  cook- 
ing rooms,  or  laundries,  D  D  eating  rooms,  S  S  sleeping  rooms,  P  P  draw- 
ing rooms,  parlors,  or  libraries,  p  p  pantries,  china  or  store  closets,  or 
clothes-presses,  c  c  water-closets  and  bath  rooms. 

Fig.  1  is  the  1st  floor  plan  of  a  very  small  square  house.  Figs.  2  and  2' 
are  1st  and  2d  floor  plans  of  a  still  larger  house.  Fig.  3  is  the  floor  plan  of  the 
same  house  differently  arranged,  the  kitchen  being  in  the  basement.  Fig. 
4,  the  same,  with  an  L  in  the  rear  for  the  kitchen.  These  plans  are  all  of 


ARCHITECTURAL    DRAWING. 


249 


Fig.  8. 


33' 


Fig.  9. 


I 


Fig.  1. 


.  1 


250  AKCHITECTURAL   DEAWIXG. 

square  houses,  and  although  not  picturesque  in  their  elevations,  are  yet 
very  convenient  and  economical  structures;  they  are  intended  for  the 
country.  The  sheds  and  back  offices  should  be  beneath  a  different  roof, 
but  attached  or  not  to  the  main  building  as  may  be  desired. 

Figs.  5,  6,  and  T,  are  first  floor  plans  of  houses  of  a  different  outline, 
but  yet  uniform,  or  nearly  so ;  in  subsequent  plates  will  be  found  illus- 
trations of  more  varied  forms  of  houses.  In  the  cities,  houses  are 
mostly  confined  to  one  form  in  their  general  outline,  a  rectangle.  Figs. 
8  and  9  may  be  taken  as  the  usual  type  of  New  York  City  houses.  Figs. 
8,  8',  8",  are  the  basement,  first,  and  second  floor  plans  of  a  Basement 
house,  three  rooms  deep.  There  is  usually  a  cellar  beneath  the  basement, 
but  in  some  cases  there  are  front  vaults,  entered  beneath  the  steps  to  the 
front  door ;  the  entrance  to  the  basement  itself  is  also  beneath  the  steps. 
The  front  room  of  the  basement  may  be  used  as  an  eating  room,  for  the 
servants'  sleeping  room,  billiards,  or  library.  The  usual  dining  room  is  on 
the  first  floor ;  a  dumb  waiter  being  placed  in  the  butler's  pantry  j?, 
for  convenience  in  transporting  dishes  to  and  from  the  kitchen.  The  ob- 
jection to  three  room  deep  houses  is  that  the  central  room  is  too  dark,  being 
lighted  by  sash  folding  doors  between  that  and  the  front  or  rear  rooms  or 
both.  Fig.  8"'  is  a  modification  to  avoid  this  objection,  the  dining  room, 
or  tea  room  as  it  is  generally  called,  being  built  as  L,  so  that  there  is  at 
least  one  window  in  the  central  room  opening  directly  out-doors.  Figs.  9, 
9',  9",  9///,  are  plans  of  the  several  floors  of  an  English  Basement  house  so 
called,  distinguished  from  the  former  in  that  the  principal  floor  is  up  one 
flight  of  stairs.  The  first  story  or  basement,  is  but  one  or  two  steps  above 
the  street,  and  contains  the  dining  room,  with  its  butler's  pantry  and  dumb 
waiter,  a  small  sitting  room,  with,  in  some  cases,  a  small  bed  room  in  the 
room  in  the  rear  of  it.  The  kitchen  is  situated  beneath  the  dining  room, 
in  the  sub-basement.  The  grade  of  the  yard  is  in  general  some  few  steps 
above  the  floor  of  the  kitchen.  Vaults  for  coal  and  provisions  are  exca- 
vated either  beneath  the  pavement  in  front  or  beneath  the  yard.  The 
advantages  of  this  form  of  house  are  the  small  sitting  room  on  the  first 
floor,  which  in  small  families,  and  in  the  winter  months,  is  the  most  fre- 
quently occupied  of  any  in  the  house ;  the  spaciousness  of  its  dining  room 
and  parlors  in  proportion  to  the  width  of  the  house,  which  is  often  but  16 
feet  8  inches  in  width,  or  three  houses  to  two  lots,  and  not  unfrequently  of 
even  a  less  width.  The  objections  to  the  house  are  the  stairs,  which  it  is 
necessary  to  traverse  in  passing  from  the  dining  rooms  or  kitchen  to  the 
sleeping  rooms,  but  this  objection  would,  of  course,  lie  against  any  house 
of  narrow  dimensions,  where  floor  space  is  supplied  by  height. 


ARCHITECTURAL    DKAWLNGt 


251 


MOULDINGS. 


It  will  have  been  observed  that  for  the  finish  of  most  of  the  parts  of  an 
edifice,  mouldings  are  found  necessary ;  so  much  so,  that  they  should 
be  classed  among  useful  rather  than  ornamental  members.  These  mould- 
ings are  drawn  either  directly  or  indirectly  from  the  Grecian  orders  of 
architecture. 

The  regular  mouldings  are  eight  in  number :  Fillet  or  Band,  Torus, 
Astragal  or  Bead,  Ovolo,  Cavetto,  Cyma  Recta  or  Ogee,  Cyma  Reversa  or 
Talon,  Scotia. 

To  construct  a  Fillet. — Thejillet,  a  (fig.  68),  is  the  smallest  rectangular 
member  employed  in  any  composition  of  mouldings.  When  it  stands  on 
a  flat  surface,  its  projection  is  usually  made  equal  to  its  height.  It  is  em- 
ployed to  separate  members. 

To  describe  a  Torus,  or  an  Astragal. — The 
torus  and  astragal  are  semicircles  in  form  pro- 
jecting from  vertical  diameters,  as  in  fig.  69.  Fig.es.  Fig.  69. 
Bisect  the  vertical  diameter  aft,  on  which  the  figure  is  projected;  on  the 
centre  <?,  describe  a  semicircle  with  c  a  as  radius.  The  astragal  is  described 
like  the  torus,  and  is  distinguished  from  it  in  the  same  order  by  being 
made  smaller.  The  torus  is  generally  employed  in  the  bases  of  columns  ; 
the  astragal,  in  both  the  base  and  capital. 

To  describe  an  Ovolo. — The  ovolo  is  a  member  strong  at  the  extremity, 
and  intended  to  support.  The  Roman  ovolo  consists  of  a  quadrant  or  a 
less  portion  of  a  circle ;  the  Greek  ovolo  is  elliptic. 

First,  the  Roman  ovolo.     When  the  projection  is  equal  to  the  height. 
Draw  a  b  for  the  height,  and  b  c  at  right  angles 
and  equal  to  it,  for  the  projection^    On  the 
centre  b  describe  the  quadrant  ca. 

When  the  projection  is  less  than  the  height. 
Draw  a  b  and  b  c  (fig.  71),  as  before,  equal  to          Fis- 70-  Fi?- "• 

the  height  and  the  projection.    On  centres  a  and  c,  with  radius  ab,  describe 
arcs  cutting  at  d  /  and  on  d  with  same  radius  describe 
the  arc  a  c  to  form  the  ovolo. 

Second,  the  Greek  ovolo.  Draw  df  from  the  lower 
end  of  the  proposed  curve,  at  the  required  inclination ; 
draw  the  vertical  g  ef  to  define  the  projection,  the 
point  6  being  the  extreme  point  of  the  curve.  Draw  Fis- T2- 

e  h  parallel  to  df,  and  draw  the  vertical  d  h  &,  such  that  d  h  is  equal  to 


fc 

~  - . 


252 


ARCHITECTURAL   DRAWING. 


Fig.  73. 


Fig.  74. 


h  Jc.  Divide  e  Ji  and  ef  into  the  same  number  of  equal  parts ;  from  d 
draw  straight  lines  to  the  points  of  division  in  ef,  and  from  k  draw  lines 
to  meet  those  others  successively.  The  intersections  so  found  are  points  in 
the  curve,  which  may  be  traced  accordingly. 

To  describe  a  Cavctto. — The  cavetto  is  described  like  the  Roman  ovolo : 

by  circular  arcs,  as  shown  in 
figs.  73  and  74.  Sometimes 
it  is  composed  of  two  circular 
arcs  united  (fig.  75) ;  set  off 
be,  two- thirds  of  the  projec- 
tion, draw  the  vertical  b  d 
equal  to  b  e,  and  on  d  describe  the  arc  b  i.  Join  e  d  and  produce  it  to  p  / 
draw  in  perpendicular  to  ed,  set  off  no  equal  to  ni,  and  draw  the 
horizontal  line  op  meeting  ep ,  on  p  describe  the  arc  io  to  complete  the 
curve. 

To  describe  a  Cyma  recta,  or  Ogee. — The  ogee  (fig.  76),  is  compounded 
of  a  concave  and  a  convex  surface.  Join  a 
and  b,  the  extremities  of  the  curve,  and  bisect 
a  b  at  c  j  on  a,  c,  as  centres,  with  the  radius 
a  c,  describe  arcs  cutting  at  d  j  and  on  b,  c, 
Fig- 76-  Fig- 77-  describe  arcs  cutting  at  e.  On  d  and  e,  as 

centres,  describe  the  arcs  a  c,  c  b,  composing  the  moulding. 

To  describe  a  Cyma  reversa,  or  Talon. — The  talon  (fig.  77),  like  the 
ogee,  is  a  compound  curve,  and  is  distinguished  from  the  other  by  having 
•    ei.-~: ;--T~~l    the  convex  part  uppermost.      It  is  described  in  the 
same  manner  as  the  ogee. 

Note. — If  the  curve  be  required  to  be  made  quicker, 
a  shorter  radius  than  ac  must  be  employed.  The  projec- 
tion of  the  moulding  n  b  (fig.  76),  is  usually  equal  to  the 
height  a  n. 

Second,  the  Greek  talon.  Join  the  extreme  points  a,  b  (fig.  78) ;  bisect 
a  b  at  c,  and  on  ac,  c  b,  describe  semicircles.  Draw 
perpendiculars  do,  &c.,  from  a  number  of  points  in 
ac,  cb,  meeting  the  circumferences ;  and  from  the  same 
points  set  off  horizontal  lines  equal  to  the  respective 
perpendiculars  :  o  n  equal  to  o  d  for  example.  The 
curve  line  b  n  a,  traced  through  the  ends  of  the  lines, 
will  be  the  contour  of  the  moulding. 

To  describe  a  Scotia. — Divide  the  perpendicular  a  b 
Fig.  79.  (fig.  79^  into  three  equal  parts ;  and  with  the  first,  a  e, 


ris- T8- 


ARCHITECTURAL   DRAWING. 


as  radius,  and  on  centre  e,  describe  the  arc  aflij  on  the  perpendicular  co, 
set  off  cl  equal  to  ae,  join  el  and  bisect  it  by  the  perpendicular  0  £?, 
meeting  c  o  at  o.  On  centre  0,  with  radius  0  c,  describe  the  arc  c  h  to 
complete  the  figure. 


ORDERS     OF     ARCHITECTURE. 

Order,  in  architecture,  is  a  system  or  assemblage  of  parts  subject  to 
certain  uniform  established  proportions,  regulated  by  the  office  each  part 
has  to  perform.  An  order  may  be  said  to  be  the  genus,  whereof  the  spe- 
cies are  Tuscan,  Doric,  Ionic,  Corinthian,  and  Composite ;  and  consists  of 
two  essential  parts :  a  column  and  entablature. 

These  are  subdivided,  the  first  into  three  parts,  namely :  the  base,  the 
shaft,  and  the  capital.  The  second  also  into  three  parts,  namely :  the 
architrave,  or  chief  beam,  C,  Plate  LIL,  which  stands  immediately  on 
the  column ;  the  frieze  B,  which  lies  on  the  architrave  ;  and  the  cornice  A, 
which  is  the  crowning  or  uppermost  member  of  an  order.  In  the  subdi- 
visions certain  horizontal  members  are  used,  which  from  the  curved  form 
of  their  edges  are  called  mouldings,  the  construction  of  which  has  already 
been  explained,  and  their  application  may  be  seen  on  the  Plate  ;  thus  a  is 
the  ogee,  5  the  corona,  c  the  ovolo,  d  the  cavetto,  which  with  fillets  com- 
pose the  cornice,yy  the  fasciae.  The  capital  of  the  column  consists  of  the 
upper  member  or  abacus  g,  the  ovolo  moulding  c,  the  astragal  ii,  and 
the  neck  h.  The  base  consists  of  the  torus  &,  and  the  plinth  I.  The 
character  of  an  order  is  displayed,  not  only  in  its  column,  but  in  its  general 
forms  and  detail,  whereof  the  column  is,  as  it  were,  the  regulator ;  the  ex- 
pression being  of  strength,  grace,  elegance,  lightness,  or  richness.  Though 
a  building  be  without  columns,  it  is  nevertheless  said  to  be  of  an  order, 
if  its  details  be  regulated  according  to  the  method  prescribed  for  such 
order. 

In  all  the  orders  a  similar  unit  of  reference  is  adopted  for  the  construc- 
tion of  their  various  parts.  Thus,  the  lower  diameter  of  the  column  is 
taken  as  the  proportional  measure  for  all  other  parts  and  members,  for 
which  purpose  it  is  subdivided  into  sixty  parts,  called  minutes,  or  into  two 
modules  of  thirty  minutes  each.  Being  proportional  measures,  modules 
and  minutes  are  not  fixed  ones  like  feet  and  inches,  but  are  variable  as  to 
the  actual  dimensions  which  they  express — larger  or  smaller,  according  to 
the  actual  size  of  the  diameter  of  the  column.  For  instance,  if  the  diam- 
eter be  just  five  feet,  a  minute,  being  one-sixtieth,  will  be  exactly  one  inch. 


254:  ARCHITECTURAL    DRAWING. 

Therefore  before  commencing  to  draw  an  elevation  of  any  one  of  the 
orders,  determine  the  diameter  of  the  column,  and  from  that  form  a  scale 
of  equal  parts,  by  sixty  divisions,  and  then  lay  off  the  widths  and  heights 
of  the  different  members  according  to  the  proportions  of  the  required 
order  as  marked  in  the  body  or  on  the  sides  of  the  plates. 

Plate  LIT.,  presents  an  illustration  of  the  Tuscan  order,  considered  by 
architects  as  a  spurious  or  plain  sort  of  Doric,  and  hardly  entitled  to  re- 
mark as  a  distinct  order,  e,  in  the  frieze  corresponding  to  the  triglyph, 
illustrates  still  further  the  connection  of  the  two  orders ;  but  by  many 
architects  this  member  is  not  introduced.  Fig.  1  is  an  elevation  of  cap- 
ital and  entablature,  fig.  2  of  the  base,  and  fig.  3  of  a  detached  capital. 
Our  example  is  constructed  according  to  the  rules  given  by  Yincent 
Scamozzi. 

Examples  of  two  capitals  are  given,  differing  merely  in  the  number  of 
mouldings  in  the  abacus.  In  fact,  this  introduction  of  simple  mouldings 
is  about  the  only  variety  allowable  in  the  order.  Ornament  is  not  admit- 
ted, nor  are  the  pillars  ever  fluted. 

A  slightly  convex  curvature,  or  entasis,  is  given  in  execution  to  the 
outline  of  the  shaft  of  a  column,  by  classic  architects,  just  sufficient  to 
counteract  and  correct  its  appearance,  or  fancied  appearance,  of  curvature 
in  a  contrary  direction  (i.  e.,  concavely),  which  might  else  take  place,  and 
cause  the  middle  of  the  shaft  to  appear  thinner  than  it  really  is. 

Fig.  4:  represents  the  form  of  a  half  column  from  the  Pantheon  at 
Home.  In  fig.  5,  another  example  of  entasis,  the  lower  third  of  the  shaft 
is  uniformly  cylindrical ;  the  two  upper  thirds  are  divided  into  seven  equal 
parts.  On  the  semicircle  shown  in  the  figure,  is  a  chord  cut  off  parallel 
to  the  diameter,  the  length  of  which  is  fifty-two  parts,  only  one-half  being 
shown.  Divide  that  part,  a  J,  of  the  circumference  between  the  diameter 
and  chord  into  seven  equal  parts,  and  draw  parallel  lines  from  each  division 
to  those  of  the  upper  part  of  the  column,  which  will  give  the  diameter  of 
the  shaft  at  each  division ;  by  increasing  the  number  of  the  divisions, 
more  diameters  for  different  parts  of  the  shaft  may  be  found. 

PI.  LIII.  exhibits  an  example  of  the  Doric  order,  from  the  temple  of 
Minerva  in  the  Island  of  Egina.  The  dimensions  are  given  in  parts  of  the 
diameter,  as  in  the  preceding  plate,  and  the  same  capital  letters  denote 
corresponding  parts.  Fig.  1  is  an  elevation  of  the  capital  and  the  entab- 
lature. Fig.  2  of  the  base,  and  a  part  of  the  Podium.  Fig.  3  shows  the 
forms  of  the  flutes  at  the  top  of  the  shaft,  and  fig.  4  at  the  base.  Fig.  5, 
the  outline  of  the  capital  on  an  enlarged  scale. 

The  Doric  order  may  be  said  to  be  the  original  of  the  Greek  orders, 


ARCHITECTURAL    DRAWING.  255 

of  winch  there  are  properly  but  three :  the  Doric,  Ionic,  and  Corinthian, 
which  differ  in  the  proportion  of  their  parts,  and  in  some  of  the  ornaments 
or  mouldings.  Of  the  Doric,  the  mutules  a  a,  the  triglyphs  b  b,  the  guttoe 
or  drops  d  d  of  the  entablature,  the  echinus/",  and  the  annulets  g  g  of  the 
capital,  may  be  considered  characteristic.  With  regard  to  the  arrangement 
of  the  triglyphs,  one  is  placed  over  every  column,  and  one  or  more  inter- 
mediately over  every  intercolumn  (or  span  between  two  columns),  at  such 
a  distance  from  each  other  that  the  metopes  c,  or  spaces  between  the  tri- 
glyphs, are  square. 

In  the  best  Greek  examples  of  the  order,  there  is  only  a  single  triglyph 
over  each  intercolumn.  One  peculiarity  of  the  Grecian  Doric  frieze  is, 
that  the  end  triglyphs,  instead  of  being,  like  the  others,  in  the  same  axis 
or  central  line  as  the  columns  beneath,  are  placed  quite  up  to  the  edge  or 
outer  angle  of  the  frieze.  The  mutules  are  thin  plates  or  shallow  blocks 
attached  to  the  under  side  or  soffit  of  the  corona,  over  each  triglyph  and 
each  metope,  with  the  former  of  which  they  correspond  in  breadth,  and 
their  soffits  or  under-surfaces  are  Avrought  into  three  rows  of  guttse  or 
drops,  conical  or  otherwise  shaped,  each  row  consisting  of  six  guttse,  or  the 
same  number  as  those  beneath  each  triglyph.  Though  a  few  exceptions  to 
the  contrary  exist,  the  shaft  of  the  Doric  column  was  generally  what  is 
technically  called  fluted.  The  number  of  channels  is  either  sixteen  or 
twenty,  afterwards  increased  in  the  other  orders  to  twenty-four ;  for  they 
are  invariably  of  an  even  number,  capable  of  being  divided  by  four ;  so 
that  there  shall  always  be  a  centre  flute  on  each  side  of  the  column. 

PI.  LIY.  presents  an  example  of  the  Ionic  order,  taken  from  the  temple 
of  Minerva  Polias  at  Athens.  Fig.  1  is  an  elevation  of  capital  and  entab- 
lature, fig.  2  of  the  base,  fig.  3  is  a  half  of  the  plan  of  the  column  at 
the  base  and  the  top,  fig.  4  an  elevation  of  the  side  of  the  capital.  In  the 
proportions  of  its  shaft,  which  are  more  slender,  and  the  addition  of  a  base, 
it  differs  from  the  Doric  ;  but  the  capital  is  the  indicial  mark  of  the  order, 
by  which  it  is  immediately  recognized.  It  is  far  more  complex  and  irreg- 
ular than  the  other  orders  of  capitals ;  instead  of  showing  four  equal  sides, 
it  exhibits  two  fronts,  with  spirals  or  volutes  parallel  to  the  architrave,  and 
narrower  baluster  sides  (fig.  4),  as  they  are  termed,  beneath  the  architrave. 

When  a  colonnade  was  continued  in  front  and  along  the  flanks  of  the 
building,  this  form  of  capital  in  the  end  column  occasioned  an  offensive 
irregularity ;  for  while  all  the  other  columns  on  the  flanks  showed  the  vo- 
lutes, the  end  one  showed  the  baluster  side.  It  was  necessary  that  the 
end  column  should,  therefore,  have  two  adjoining  volute  faces,  which  was 
effected  by  placing  the  volute  at  the  angle  diagonally,  so  as  to  obtain  there 


256  ARCHITECTURAL   DRAWING. 

two  voluted  surfaces  placed  immediately  back  to  back.  This  same  diag- 
onal disposition  of  the  volutes  is  employed  for  all  the  capitals  alike,  in 
Koman  and  Italian  examples  of  this  order. 

The  capital  admits  of  great  diversity  of  character  and  decoration — it 
sometimes  is  without  necking,  sometimes  with ;  which  may  either  be  plain 
or  decorated,  to  suit  the  entire  design.  The  capital  may  also  be  modified 
in  its  proportions,  first  as  regards  its  general  proportion  to  the  column ; 
secondly,  as  regards  the  size  of  the  volutes  compared  with  the  width  of 
the  face.  In  the  best  Greek  examples,  the  volutes  are  much  bolder  than 
in  the  Roman.  The  spirals  also  of  the  volutes  may  be  either  single  or 
manifold,  and  the  eye  or  centre  of  the  spiral  may  be  made  larger  or  smaller, 
flat  or  convex,  or  curved  as  a  rosette. 

Plate  LY.  represents  an  example  of  the  Corinthian  order,  from  the  Arch 
of  Hadrian,  at  Athens.  This  order  is  distinguished  from  the  Ionic,  more 
by  its  deep  and  foliaged  capital  than  by  its  proportions, — the  columns  of 
both  have  bases  differing  but  little  from  each  other,  and  their  shafts  are 
fluted  in  the  same  manner. 

Although  the  order  itself  is  the  most  delicate  and  lightest  of  the  three, 
the  capital  is  the  largest,  being  considerably  more  than  a  diameter  in 
height,  varying  in  different  examples  from  one  to  one  and  a  half  diame- 
ter, upon  the  average  about  a  diameter  and  a  quarter. 

The  capital  has  two  rows  of  leaves,  eight  in  each  row,  so  disposed  that 
of  the  taller  ones,  composing  the  upper  row,  one  comes  in  the  middle,  be- 
neath each  face  of  the  abacus,  and  the  lower  leaves  alternate  with  the 
upper  ones,  coming  between  the  stems  of  the  latter ;  so  that  in  the  first 
or  lower  tier  of  leaves  there  is  in  the  middle  of  each  face,  a  space  between 
two  leaves  occupied  by  the  stem  of  the  central  leaf  above  them.  Over 
these  two  rows  is  a  third  series  of  eight  leaves,  turned  so  as  to  support  the 
small  volutes  which,  in  turn,  support  the  angles  of  the  abacus.  Besides 
these  outer  volutes,  which  are  invariably  turned  diagonally,  as  in  the 
y  /  four-faced  Ionic  capital,  there  are  two  other 

smaller  ones,  termed  caulicoU,  which  meet  each 
other  beneath  a  flower  on  the  face  of  the  abacus. 
The  abacus  itself  is  not,  properly  speaking,  a 
square,  although  it  may  be  said  to  be  so  in  its 
general  form.  But  instead  of  being  straight, 
the  sides  of  the  abacus  are  concave  in  plan,  be- 
ing curved  outwards  so  as  to  produce  a  sharp 


Fig  ^  point  at  each  corner,  which  is  usually  cut  off. 

Fig.  80  represents  one  of  the  capitals  of  the  Tower  of  the  Winds,  show- 


ARCHITECTURAL   DRAWING.  257 

ing  the  earliest  formation  of  the  Corinthian  capital.  In  this  example  the 
abacus  is  square,  and  the  upper  row  of  leaves  of  the  kind  called  water 
leaves,  from  their  resemblance  to  those  of  water  plants,  being  broad  and 
flat,  and  merely  carved  upon  the  vase  or  body  of  the  capital. 

The  proper  Corinthian  base  differs  from  that  of  the  usual  Ionic  or  Attic, 
in  having  two  smaller  scotise,  separated  by  two  astragals :  however,  both 
kinds  are  employed  indiscriminately.  The  shaft  is  fluted,  in  general,  simi- 
larly to  that  of  the  Ionic  column,  but  sometimes  the  flutes  are  cabled  as  it 
is  called,  that  is,  the  channels  are  hollowed  out  for  only  about  two-thirds 
of  the  upper  part  of  the  shaft,  and  the  remainder  cut  so  that  each  channel 
has  the  appearance  of  being  partly  filled  up  by  a  round  staff  or  piece  of 
rope,  whence  the  term  cabling. 

The  cornice  is  very  much  larger  than  in  the  other  orders, — larger  as  to 
height,  and  consequently  as  to  projection  also. 

From  this  greatly  increased  depth  of  cornice,  it  consists  of  a  greater 
number  of  mouldings  beneath  the  corona,  for  that  and  the  cyinatium  over 
it  invariably  retain  their  places  as  the  crowning  members  of  the  whole 
series  of  mouldings.  In  our  illustration,  square  blocks  or  dentels  are 
introduced,  but  often  to  the  dentels  is  added  a  row  of  modillions^  imme- 
diately beneath  and  supporting  the  corona.  These  modillions  are  orna- 
mental blocks,  curved  in  their  under  surface  somewhat  after  the  manner 
of  the  letter  S  turned  thus,  QQ  ;  and  between  them  and  the  dentels,  and 
also  below  the  latter,  are  ether  mouldings,  sometimes  cut,  at  others  left 
plain.  Sometimes  a  plain  uncut  dentel  land  is  substituted  for  dentels ; 
sometimes,  in  simpler  cornices,  that  is  omitted  altogether,  and  plainer 
blocks  are  employed  instead  of  modillions ;  or  else  both  dentels  and  mo- 
dillions are  omitted.  The  dentel  is  not  peculiar  to  this  order,  but  is  con- 
sidered as  more  properly  belonging  to  the  Ionic. 

The  Composite  Order  is  hardly  to  be  considered  as  a  distinct  order, 
being  but  a  union  of  the  Ionic  and  Corinthian.  Its  capital  consists  of  a 
Roman  Ionic  one,  super-imposed  upon  a  Corinthian  foliaged  base,  in  which 
the  leaves  are  without  stalks,  placed  directly  upon  the  body  of  the  vase. 
In  general,  the  entablature  is  Corinthian — but  in  a  few  examples  it  is  Ionic. 

Although  columns  and  entablatures  do  not  of  themselves,  properly 
speaking,  constitute  an  order,  except  they  enter  into  the  organization  of  a 
structure ;  yet,  as  the  Greek  edifices  as  such,  are  almost  entirely  inappli- 
cable to  purposes  of  the  present  day,  we  have  confined  our  illustrations 
of  the  orders  to  the  pillars  and  entablatures  merely,  remarking,  that  how- 
ever the  Greek  temples  differed  from  each  other  as  to  the  treatment  of  the 
order  adopted,  the  number  of  columns  and  mere  particulars  of  that  kind, 
17 


258  ARCHITECTURAL,    DRAWING. 

they  resemble  each  other.'  Not  only  were  their  plans  invariably  parallel- 
ograms, but  alike  also  as  to  proportion,  forming  a  double  square,  or  being 
about  twice  as  much  in  length  as  in  breadth.  The  number  of  the  columns 
in  front  was  invariably  an  even  one,  so  that  their  might  be  a  central  inter- 
column  ;  but  on  the  flanks  of  the  edifice,  where  there  was  no  entrance,  the 
number  of  intercolumns  was  an  even,  and  that  of  the  columns  an  uneven 
one,  so  that  a  column  came  in  the  centre  of  these  side  elevations. 

As  to  the  mode  in  which  the  front  influenced  the  sides,  by  determining 
the  number  of  columns  for  them,  the  established  rule  seems  to  have  been 
to  give  the  flanks  twice  as  many  intercolumns  as  there  were  columns  at 
each  end :  thus  the  Parthenon,  which  is  octastyle  or  eight  columns  in  front, 
has  sixteen  intercolumns,  and  consequently  seventeen  columns,  on  each 
flank.  In  like  manner,  a  hexastyle  temple  would  have  twelve  intercol- 
umns and  thirteen  columns  on  each  side. 

In  the  Doric  order,  the  distances  between  the  columns  is  governed 
entirely  by  the  triglyphs  of  the  frieze,  so  that  there  can  be  no  medium 
between  monotriglyphic  and  ditriglyphic  intercolumniation,  accordingly 
as  there  is  either  one  or  two  triglyphs  over  each  intercolumn.  But  in  the 
other  orders  there  is  no  such  restriction ;  in  them  the  intercolumns  may  be 
made  wider  or  narrower,  as  circumstances  require,  from  one  diameter  and  a 
quarter  to  a  half  in  width.  Close  spacing  carries  with  it  the  expression  of 
both  richness  and  strength,  whilst  wide  spacing  produces  an  effect  of  open- 
ness and  lightness,  but  also  partakes  of  meagreness  and  weakness,  owing 
to  the  want  of  sufficient  apparent  support  for  the  entablature. 

Another  mode  of  columniation  and  intercolumniation  which  has  some- 
times been  practised  by  Modern  Architects,  consists  in  coupling  the 
columns  and  making  a  wide  intercolumn  between  every  pair  of  columns, 
so  that  as  regards  the  average  proportion  between  solids  and  voids,  that 
disposition  does  not  differ  from  what  it  would  be  were  the  columns 
placed  singly.  Supercolimmiation,  or  the  system  of  piling  up  orders,  or 
different  stages  of  columns  one  above  another,  was  employed  for  such 
structures  merely  as  were  upon  too  large  a  scale  to  admit  of  the  applica- 
tion of  columns  at  all  as  their  decoration,  otherwise  than  by  disposing  them 
in  tiers.  This  method  was  afterwards  adopted  by  the  Architects  of  the  Pal- 
ladian  School.  Sometimes  all  the  three  orders  are  employed  in  as  many 
tiers  of  columns  or  pilasters.  In  other  cases,  the  two  extreme  orders, — 
that  is,  the  Doric  and  Corinthian, — are  brought  together ;  in  other  cases 
but  a  single  order. 

In  one  or  two  instances  the  Greeks  employed  human  figures  to  support 
entablatures  or  beams ;  the  female  figures  or  Caryatides,  are  almost  uni- 


AECniTECTUEAL   DRAWING.  259 

formly  represented  in  an  erect  attitude,  without  any  apparent  effort  to 
sustain  any  burden  or  load ;  whilst  the  male  figures,  Telamones  or  Atlantes, 
manifestly  display  strength  and  muscular  action.  Besides  entire  figures, 
either  Hermes'  pillars  or  Termini  are  occasionally  used  as  substitutes  for 
columns  of  the  usual  form,  when  required  to  be  only  on  a  small,  at  least  a 
moderate  scale.  The  first  mentioned  consist  of  a  square  shaft  with  a  bust 
or  human  head  for  its  capital ;  the  latter  of  a  half-length  figure  rising  out 
of,  or  terminating  in,  a  square  shaft  tapering  downwards.  Hermes'  pillars 
seem  to  be  in  great  favor  with  modern  German  architects,  they  having  not 
unfrequently  employed  them  for  the  decoration  of  windows. 

The  Greek  orders  may  be  considered  as  the  rudiments  of  modern 
architecture,  but  the  forms  of  their  buildings  are  almost  entirely  inappli- 
cable to  modern  purposes.  The  Eomans  developed  and  matured  the 
Corinthian  order,  and  also  worked  out  a  freer  and  more  complex  and  com- 
prehensive system  of  architecture.  They  introduced  circular  forms  and 
curves  not  only  in  elevation  and  section,  but  in  plan ;  and  while,  among 
the  Greeks,  architecture  was  confined  almost  exclusively  to  external  ap- 
pearance and  effect,  in  the  hands  of  the  Romans  it  was  made  to  minister 
to  internal  display  also.  The  true  Roman  order  consists,  not  in  any  of  the 
columnar  ordinances,  but  in  an  arrangement  of  two  pillars  placed  at  a  dis- 
tance from  one  another  nearly  equal  to  their  own  height,  and  having  a 
very  long  entablature,  which,  in  consequence,  required  to  be  supported  in 
the  centre  by  an  arch  springing  from  piers.  This,  as  will  be  seen  from 
fig.  1,  Plate  LVI.  was,  in  fact,  merely  a  screen  of  Grecian  architecture 
placed  in  front  of  an  arcade.  Though  not  without  a  certain  richness  of 
effect,  still  as  used  by  the  Romans,  these  two  systems  remain  too  distinctly 
dissimilar  for  the  result  to  be  pleasing,  and  their  use  necessitated  certain 
supplemental  arrangements  by  no  means  agreeable.  In  the  first  place  the 
columns  had  to  be  mounted  on  pedestals,  or  otherwise  an  entablature  pro- 
portional to  their  size  would  have  been  too  heavy  and  too  important  for  a 
thing  so  useless  and  so  avowedly  a- mere  ornament.  A  projecting  key- 
stone was  also  introduced  into  the  arch.  This  was  unobjectionable  in 
itself,  but  when  projecting  so  far  as  to  do  the  duty  of  an  intermediate  cap- 
ital, it  overpowered  the  arch  without  being  equal  to  the  work  required  of 
it.  The  Romans  used  these  arcades  with  all  the  three  orders,  frequently 
one  over  the  other,  and  tried  various  expedients  to  harmonize  the  construc- 
tion with  the  ornamentation,  but  without  much  effect.  They  seem  always 
to  have  felt  the  discordance  as  a  blemish,  and  at  last  got  rid  of  it,  remov- 
ing the  pier  altogether,  and  substituting  in  its  place  the  pillar  taken  down 
from  its  pedestal.  This  of  course  was  not  effected  at  once,  but  was  the 


260  ARCHITECTURAL   DRAWING. 

result  of  many  trials  and  expedients.  One  of  the  earliest  of  these  is  ob- 
served in  the  Ionic  Temple  of  Concord,  in  which  a  concealed  arch  is 
thrown  from  the  head  of  each  pillar,  but  above  the  entablature,  so  as  to 
take  the  whole  weight  of  the  superstructure  from  off  the  cornice  between 
the  pillars.  When  once  this  was  done  it  was  perceived  that  so  deep  an 
entablature  was  no  longer  required,  and  that  it  might  be  either  wholly 
omitted,  as  was  sometimes  done,  in  the  centre  intercolumniation,  or  at  all 
events  very  much  attenuated.  There  is  an  old  temple  at  Talavera,  in 
Spain,  which  is  a  good  example  of  the  former  expedient ;  and  the  Church 
of  the  Holy  Sepulchre,  built  by  Constantino  at  Jerusalem,  is  a  remarkable 
example  of  the  latter.  There,  the  architrave  is  cut  off  so  as  merely  to 
form  a  block  over  each  of  the  pillars,  and  the  frieze  and  cornice  only  are 
carried  across  from  one  of  these  blocks  to  the  other,  while  a  bold  arch  is 
thrown  from  pillar  to  pillar  over  these,  so  as  to  take  any  weight  from  off  a 
member  which  has  at  last  become  a  mere  ornamental  part  of  the  style. 

Figs.  2,  3  and  4,  Plate  LYI.  from  the  Palace  of  Diocletian  at  Spalatro, 
are  illustrations  of  the  different  modes  of  treatment  of  the  arch  and  entab- 
lature. 

Perhaps  the  most  satisfactory  works  of  the  Romans  are  those  which  we 
consider  as  belonging  to  civil  engineering  rather  than  to  architecture; 
their  aqueducts  and  viaducts,  all  of  which,  admirably  conceived  and  exe- 
cuted, have  furnished  practical  examples  for  modern  constructions,  of  which 
the  High  Bridge  across  Harlem  River  may  be  taken  as  an  illustration. 

The  whole  history  of  Roman  architecture  is  that  of  a  style  in  course 
of  transition,  beginning  with  purely  Pagan  or  Grecian,  and  passing  into  a 
style  almost  wholly  Christian.  The  first  form  which  Christian  art  took  in 
emancipating  itself  from  the  Pagan  was  the  Romanesque,  which  afterwards 
branched  off  into  the  Byzantine  and  the  Gothic. 

The  Romanesque  and  Byzantine,  as  far  as  regards  the  architectural 
features,  are  almost  synoymous;  in  the  earlier  centuries  there  is  an  orna- 
mental distinction,  the  Romanesque  being  simply  a  debasement  of  Roman 
art — the  Byzantine  being  the  art  combined  with  the  symbolic  elements 
introduced  by  the  new  Christian  religion.  As  commonly  used,  the  dome 
is  also  considered  a  characteristic  of  the  Byzantine,  but  this  will  be  found 
among  Roman  examples.  In  its  widest  signification,  the  Romanesque  is 
applied  to  all  the  earlier  round  arch  developments,  in  contradistinction  to 
the  Gothic  or  later  pointed  arch  varieties  of  the  North.  In  this  view  the 
Norman  is  included  in  the  Romanesque,  and  this  distinction  will  be  suffi- 
cient for  our  purpose. 

The  general  characteristics  of  the  Gothic  are  these :  it  is  essentially 


ARCHITECTURAL   DRAWING.  261 

pointed  or  vertical  in  its  tendency,  in  its  details  geometrical,  in  its  win- 
dow tracery,  in  its  openings,  in  its  cluster  of  shafts  and  bases,  in  its  suits  of 
mouldings,  and  by  the  universal  absence  of  the  dome,  and  the  substitution 
of  the  pointed  for  the  round  arch. 

The  Romanesque  pillars  are  mostly  round  or  square,  and  if  square, 
generally  set  evenly,  whilst  the  Gothic  square  pillar  is  set  diagonally. 

Figs.  5,  6,  7,  8  and  9,  Plate  LYI.,  represent  sections  of  Gothic  pillars  ; 
fig.  10  is  half  of  one  of  the  great  western  piers  of  the  Cathedral  of  Bourges, 
measuring  8  feet  on  each  side. 

Figs.  11  and  12  are  the  elevations  of  capitals  and  bases  and  the  sec- 
tions of  Gothic  pillars,  one  from  Salisbury,  the  other  from  Lincoln  Cathe- 
dral. Fig.  13  is  a  Byzantine  capital  from  the  church  of  St.  Sophia  at 
Constantinople ;  fig.  14  one  from  the  palace  at  Gelnhausen ;  fig.  15,  a  Nor- 
man one,  from  Winchester  Cathedral,  and  fig.  16  a  Gothic  capital  and  base 
from  Lincoln  Cathedral. 

Mouldings. — "  All  classical  architecture,  and  the  Eomanesque  which  is 
legitimately  descended  from  it,  is  composed  of  bold  independent  shafts, 
plain  or  fluted,  with  bold  detached  capitals  forming  arcades  or  colonnades 
where  they  arc  needed,  and  of  walls  whose  apertures  are  surrounded  by 
courses  of  parallel  lines  called  mouldings,  and  have  neither  shafts  nor 
capitals.  The  shaft  system  and  moulding  system  are  entirely  separate,  the 
Gothic  architects  confounded  the  two ;  they  clustered  the  shafts  till  they 
looked  like  a  group  of  mouldings,  they  shod  and  capitalled  the  mouldings 
till  they  looked  like  a  group  of  shafts." 

Gothic  Mouldings  appear  in  almost  every  conceivable  position ;  from 
the  bases  of  piers  and  piers  themselves,  to  the  ribs  of  the  fretted  vaults 
which  they  sustain,  scarce  a  member  occurs  which  is  incapable  of  receiv- 
ing consistent  decoration  by  this  elegant  method. 

Jamb  Mouldings. — In  the  earliest  examples  of  Nor- 
man doorways,  the  jambs  are  mostly  simply  squared 
back  from  the  walls;  recessed  jambs  succeeded,  and  are 
common  in  both  Norman  and  Gothic  architecture ;  and 
when  thus  raised  detached  shafts  were  placed  in  each 
angle  (fig.  81).  In  the  later  styles,  the  shafts  were  almost 
invariably  attached  to  the  structure.  The  angles  them-  Fig.  si.  * 

selves  were  often  cut  or  chamfered  ofF,  and  the  mouldings  attached  to  the 
chamfer  plane.  The  arrangement  of  window  jambs  during  the  successive 
periods  was  in  close  accordance  with  that  of  doorways. 

In  the  richer  examples  small  shafts  were  introduced,  which,  rising  up  to 
the  springing  of  the  window,  carried  one  or  several  of  the  arch  mouldings. 


262 


ARCHITECTURAL   DRAWIXG. 


Yet  mouldings  are  not  nevertheless  essential  accessories ;  many  windows 
of  the  richest  tracery  have  their  nmllions  and  jambs  composed  of  simple 
chamfers. 

Arch  Mouldings,  even  when  not  continuous,  partook  of  the  same 
general  arrangement  as  those  in  the  jambs,  with  greater  richness  of  detail. 
"When  shafts  were  employed,  they  carried  groups  of  mouldings  more 
elaborate  than  those  of  the  jambs,  though  still  falling  on  the  same  planes. 
Capitals  were  either  moulded,  or  carved  with  foliage,  animals,  &c. ; 
they  always  consisted  of  three  distinct  parts  (fig.  82),  the 
head  mould  A,  the  bell  B,  and  the  neck  mould  C.     In 
Norman  examples  the  head  mould  was  almost  invariably 
square  ;  in  the  later  styles  it  is  circular,  or  corresponding 
to  the  form  of  the  pillar. 

Bases  consist  of  the  plinth  and  the  base  mouldings. 
The  plinth  was  square  in  the  Norman  style,  afterwards 
octagonal,  then  assuming  the  form  of  the  base  mouldings, 
it  bent  in  and  out  with  the  outline  of  the  pier.  Base  mouldings  were  also 
extensively  used  round  the  buttresses,  towers  and  walls  of  churches. 

String  Courses. — The  most  usual  and\  perhaps  essential  position  of  the 
string  course  is  under  the  windows.  Iti  the  Norman  styles  they  were 
usually  heavy  in  the  outline,  and  displayeq  no  particular  beauty  of  arrange- 
ment. In  the  later  styles  they  were  remarkably  light  and  elegant ;  from 
restraint  or  horizontally,  they  now  rose  close  under  the  sill  of  the  window, 
and  then  suddenly  dropping  to  accommodate  themselves  to  the  arch  of  a 
low  doorway,  and  again  rising  to  run  immediately  under  the  adjoining  win- 
dow. In  this  way,  the  string  courses  frequently  served  the  purpose  of  a 
drip  stone  or  hood  moulding  over  doors ;  occasionally  the  hood  mould  was 
continued  from  one  window  to  the  other.  But  in  the  later  styles  they  were 
generally  terminated  in  heads,  flowers,  or  some  quaint  device,  or  simply 
returned  at  the  springing  of  the  arch. 

Cornices  are  not  an  essential  feature  in  Gothic  architecture.  In  the 
Norman  and  early  English  styles,  the  cornice  was  a  sort  of  enlarged  string 
course  formed  by  the  projection  of  the  upper  part  of  the  wall,  which  was 
supported  on  brackets  or  corbels,  and  hence  termed  the  corbel  table. 

The  earliest  moulding  in  Norman  work,  is  a  circular  bead  strip  worked 
out  of  the  edges  of  a  recessed  arch,  called  a  circular  bowtel 
(fig.  83).  From  a  circular  form  the  bowtel  soon  became 
pointed,  and,  by  an  easy  transition,  into  the  bowtel  of  one, 
two,  or  three  fillets,  all  of  which,  with  their  numerous  varieties, 
performed  important  parts  in  the  Gothic  moulding  system. 


ARCHITECTURAL   DRAWING. 


263 


Fig.  84  is  the  scroll  moulding,  being,  in  fac^  a  simple  filleted  bowtel, 
with  the  fillet  undeveloped  on  one  side,  as  shown  by  the 
dotted  lines.  If  this  moulding  be  cut  in  half,  through 
the  centre  of  the  fillet,  we  have  on  the  developed  side 
the  moulding  now  termed  by  carpenters  the  rule  joint, 
which,  by  rounding  off  the  corners  by  reverse  curves,  becomes  the  wave 
moulding. 

The  ogee  is  also  used  very  generally  in  Gothic  architecture,  both  single 
and  double,  the  latter  formed  by  the  junction  of  two  ogees. 

Figs.  85  and  86  are  examples  of 
groupings  of  mouldings,  fig.  85  being 
of  the  earlier  Gothic,  the  filleted  bow- 
tel  with  alternate  hollows,  fig.  86,  of 
the  perpendicular  style,  the  hollow  in 
the  one  case  being  made  prominent, 
and  dividing  individual  mouldings ; 
in  the  latter  insignificant,  and  as  a 
separation  of  groups  of  mouldings. 

Arches  are  generally  divided  into  the  triangular-headed  arch,  the 
round-headed  arch,  and  the  pointed  arch.  Of  round-headed  arches  there 
are  four  kinds,  the  semicircular,  segmental,  the  stilted,  and  the  horse-shoe. 

The  stilted  arch,  fig.  87,  is  semicircular,  but  the  sides  are  carried  down- 
wards in  a  straight  line  below  the  spring  of  the  curve,  till  they  rest  upon 
the  imposts.  In  the  horse-shoe  arch,  the  sides  are  also  carried  down  be- 
low the  centre,  but  follow  the  same  curve  (fig. 


Fig.  86. 


Fig-  87.  Fig.  83.  Fig.  89.  Fig.  90. 

The  pointed  arch  may  be  divided  into  two  classes,  those  described  from 
two  centres,  and  those  described  from  four.  Of  the  first  class  there  are 
three  kinds,  the  equilateral,  the  lancet,  and  the  obtuse.  The  equilateral 
(fig.  89),  is  formed  of  two  segments  of  a  circle,  of  which  the  radii  are 
equal  to  the  breadth  of  the  arch.  The  radii  of  the  lancet  segment  are 
longer  than  the  width  of  the  arch,  and  of  the  obtuse,  shorter. 

Of  the  complex  arches,  there  is  the  Ogee  (fig.  90),  and  the  Tudor  (fig. 
92).  The  Tudor  arch  is  described  from  four  centres,  two  on  a  level  with 
the  spring  and  two  below  it. 


264 


AECHITECTUEAL   DRAWING. 


Of  foiled  arches,  there  are  the  round-headed  trefoil  (fig.  91),  the  pointed 
trefoil  (fig.  93),  and  the  square-headed  trefoil  arch  (fig.  94). 


Fig.  91. 


Fig.  92. 


Fig.  93. 


Fig.  94 


The  semi-circular  arch  is  the  Roman  Byzantine  and  Korman  arch,  the 
ogee  and  horse-shoe  is  the  profile  of  many  Turkish  and  Moorish  domes,  the 
pointed  and  foliated  arches  are  Gothic.  * 

Domes  and  Vaults. — Both  domes  and  vaults  are  found  in  Eoman 
works,  but  with  the  decline  of  Roman  power  the  art  of  vaulting  was  lost, 
and  the  churches  of  all  Roman  Christendom  remained  with  nothing  but 
timber  roofs.  But  among  the  Greek  Christians,  or  Byzantines,  it  was  re- 
tained, or  else  re-invented ;  but  the  Greek  vaulting  consisted  wholly  of 
spherical  surfaces,  whilst  the  Roman  consisted  of  cylindrical  ones.  Figs. 
95  and  96  illustrate  this  distinction,  fig.  95  being  the  elevation  of  a  Roman 
cylindrical  cross  vault,  and  fig.  96,  the  elevation  of  the  roof  of  the  church 


Fig.  95. 


Fig.  96. 


of  St.  Sophia  at  Constantinople  ;  and  the  sprouting  of  domes  out  of  domes 
continues  to  characterize  the  Byzantine  style,  both  in  Greek  churches  and 
Turkish  mosques,  down  to  the  present  day.  This  system  of  vaulting  has 
also  been  adopted  in  St.  Paul's,  London,  and  at  St.  Genevieve,  Paris.  As 
a  constructive  expedient  the  -cross  vault  is  to  be  preferred,  as  the  whole 
pressure  and  thrust  are  collected  in  four  definite  resultants,  applied  at  the 
angles  only,  so  that  it  might  be  supported  by  four 
flying  buttresses,  no  matter  how  slender,  provided 
they  were  placed  in  the  direction  of  these  result- 
ants, and  were  strong  enough  not  to  be  crushed  by 
the  pressure. 

Fig.  97  represents  a  compartment  of  the  sim- 
plest Gothic  vaulting, «,  a,  groin  ribs,  J,  5, 5,  side  ribs. 
The  Romans  introduced  side  ribs,  appearing 


Fig.  97. 


ARCHITECTURAL    DRAWING.  265 

on  the  inside  as  flat  bands,  and  harmonizing  with  the  similar  form  of 
pilasters  in  the  walls,  but  they  never  nsed  groin  ribs ;  the  Gothic  build- 
ers introduced  these,  and  deepened  the  Roman  ribs.  The  impenetra- 
tion  of  vaults,  either  round  or  pointed,  produces  elliptical  groin  lines, 
or  else  lines  of  double  curvature.  Yet  the  early  Gothic  architects  rarely 
made  their  groin  ribs  elliptical,  and  never  deviating  from  a  vertical  plane. 
These  ribs  were  usually  simple  pointed  arches  of  circular  curvature, 
thrown  diagonally  across  the  space  to  be  groined,  and  the  four  side  arches 
were  equally  simple,  the  only  care  being  that  all  the  arches  should  have 
their  vertices  at  the  same  level.  The  shell  between,  therefore,  was  no 
regular  geometric  surface.  The  strength  depended  on  the  ribs,  and  the 
shell  was  made  quite  light,  often  not  more  than  six  inches,  while  Roman 
vaults  of  the  same  span  would  have  been  three  or  four  feet.  The  differ- 
ence- of  principle  being,  that  the  Romans  made  their  vault  surfaces  geo- 
metrically regular,  and  left  the  groins  to  take  their  chance ;  while  the  early 
Gothic  architects  made  their  groins  geometrically  regular,  and  let  the  in- 
termediate surfaces  take  their  chance. 

In  the  next  step  the  groin  ribs  were  elliptical,  and  when  intermediate 
ribs  or  tiercerons  were  inserted,  these  ribs  had  also  elliptical  curvatures, 
but  different  from  the  groins,  in  order  that  the  vault  of  cut  stone  built 
upon  them  might  have  a  regular  cylindrical  surface.  In  augmenting  the 
number  of  tiercerons,  and  making  them  ramify,  combinations  of  circular 
arcs  were  substituted  for  the  elliptic  curves ;  the  surfaces  of  these  vaults 
could  not  be  cylindrical,  but  the  ribs  were  placed  very  near  each  other,  in 
order  that  the  portion  of  the  vault  between  each  pair  might  practically  be 
almost  cylindrical.  In  the  formation  of  the  compound  circular  ribs  three 
conditions  were  to  be  observed : — 1st,  that  the  two  arcs  should  have  a 
common  tangent  at  the  point  of  meeting.  2d.  That  the  feet  of  all  the 
ribs  should  have  the  same  radius,  up  to  the  level  at  which  they  completely 
separate  from  each  other.  3d.  That  from  this  point  upwards,  their  curva- 
tures should  be  so  adjusted  as  to  make  them  all  meet  their  fellows  on  the 
same  horizontal  plane,  so  that  all  the  ridges  of  the  vaults  may  be  on  one 
level. 

The  geometrical  difficulty  of  such  works  led  to  what  is  called  fan 
tracery  vaulting.  If  similar  arches  spring  from  each  side  of  the  pillars 
(fig.  97),  it  is  easy  to  perceive  that  the  portion  of  vault  springing  from 
each  pillar  would  have  the  form  of  an  inverted  concave-sided  pyramid, 
its  horizontal  section  at  every  level  being  square.  Now  the  later  archi- 
tects converted  this  section  into  a  circle,  the  four-sided  pyramid  became  a 
conoid,  and  all  the  ribs  forming  the  conoidal  surface  became  alike  in  cur- 


266  ARCHITECTURAL   DRAWING. 

vature,  so  that  they  all  might  be  made  simple  circular  arcs ;  these  ribs 
are  continued  with  unaltered  curvature  till  they  meet  and  form  the 
ridge ;  but  in  this  case  the  ridges  are  not  level,  but  gradually  descend 

every  way  from  the  centre  point  (fig.  98). 

In  the  figure  this  is  not  fully  carried  out, 
for  no  rib  is  continued  higher  than  those 
over  the  longer  sides  of  the  compartment, 
so  that  a  small  lozenge  is  still  left,  with  a 
boss  at  its  centre.  When  the  span  of  the 
main  arch  1)  a,  was  large  in  proportion  to 
that  of  I  GJ  the  arch  £  c  became  a  very  acute 
'rig.  98.  lancet  arch,  and  scarcely  admitting  win- 

dows of  an  elegant  or  sufficient  size.  To  obviate  this,  the  compound  curve 
was  again  introduced,  and  the  ribs  were  made  less  curved  in  their  upper 
parts  than  in  the  lower.  Hence  the  four- centred  or  Tudor  arches. 

The  four-centred  arch  is  not  necessarily  flat  or  depressed,  it  can  be 
made  of  any  proportion,  high  or  low,  and  always  with  a  decided  angle  at 
the  vertex.  In  general,  the  angular  extent  of  the  lower  curve  is  not  more 
than  65°,  nor  less  than  45°.  The  radius  of  the  upper  curve  varies  from 
twice  to  more  than  six  times  the  radius  of  the  lower,  but  generally  speak- 
ing, the  greater  their  disproportion,  the  less  pleasing  is  the  sudden  change 
of  curvature.  The  projecting  points  of  the  trefoil  arch  are  sometimes 
called  cusps,  often  introduced  for  ornament  merely,  but  serving  construc- 
tively both  in  vaults  and  arches,  as  a  load  for  the  sides,  to  prevent  them 
rising  from  the  pressure  on  the  crown.  This  property  of  arches  has  been 
explained,  depending  on  the  principle  (p.  118),  that  if  a  polygon  of  rods 
be  reversed,  the  position  in  which  it  will  stand  is  that  which  it  will  assume 
for  itself  when  loaded  with  the  same  weights  and  suspended ;  and  perhaps 
the  equilibrium  of  some  of  the  boldest  vaultings  was  insured  by  experi- 
ments on  systems  of  rods  representing  the  ribs  inverted ;  and  for  any  archi- 
tect who  may  wish  to  introduce  pendants  or  cusps  in  his  vaultings,  this 
rule  of  trial  will  be  found  particularly  useful. 

As  vaultings,  in  general,  were  contrived  to  collect  the  whole  pressure 
of  each  compartment  into  four  single  resultants,  at  the  points  of  springing, 
leaving  the  walls  so  completely  unloaded  that  they  are  required  only  as 
enclosures  or  screens,  they  might  be  entirely  omitted  or  replaced  by  win- 
dows. Indeed,  the  real  supporting  walls  are  broken  into  narrow  slips, 
placed  at  right  angles  to  the  outline  of  the  building,  and  called  luUresses. 
As  to  the  enclosing  walls,  being  not  for  support,  they  may  be  placed  as 


AECHITECTTJKAL   DRAWING.  267 

the  architect  pleases,  either  at  the  outer  or  inner  edge  of  the  buttresses. 
The  one  method,  being  that  adopted  by  the  French  architects,  gave  to 
their  interiors  those  deep  recesses,  whilst  the  other,  or  English  method, 
served  only  to  produce  external  play  of  light  and  shade. 

The  Gorman  buttress  resembles  a  flat  pilaster,  being  a  mass  of  masonry 
with  a  broad  face,  slightly  projecting  from  the  wall.  They  are,  generally, 
of  but  one  stage,  rising  no  higher  than  the  cornice,  under  which  they 
often,  but  not  always,  finish  with  a  slope.  Sometimes  they  are  carried  up 
to,  and  terminate  in,  the  corbel  table. 

Fig.  1,  Plate  LYII.  represents  a  buttress  in  two  stages,  with  simple 
slopes  as  set-offs ;  this  example  is  somewhat  narrower  and  projects  more 
than  the  Norman  buttress. 

Fig.  2  is  a  buttress  of  the  Early  English  style,  having  a  plain  triangu- 
lar or  pedimental  head.  The  angles  were  sometimes  chamfered  off,  and 
sometimes  ornamented  with  slender  shafts.  In  buttresses  of  different 
stages,  the  triangular  head  or  gable  is  used  as  a  finish  for  the  intermediate 
stages. 

In  the  Decorated  style,  the  outer  surfaces  of  the  buttresses  are  orna- 
mented with  niches,  as  in  fig.  3.  In  the  Perpendicular  style,  the  outer 
surface  is  often  partially  or  wholly  covered  with  panel-work  tracery 
(fig.  4). 

It  has  been  said  that  the  buttress  was  a  constructive  expedient  to  re- 
sist the  thrust  of  vaulting,  but  to  resist  the  thrust  of  the  principal  vault, 
or  that  over  the  nave  or  central  part  of  the  church,  buttresses  of  the  re- 
quisite depth  would  have  filled  up  the  side  aisles  entirely.  To  obviate 
this,  the  system  of  flying  buttresses  'was  adopted,  that  is,  the  connection 
of  the  interior  with  the  outer  buttress,  by  an  arch  or  system  of  arches,  as 
shown  in  fig.  5.  To  add  weight,  and  consequently  solidity,  to  the  outer 
piers,  they  were  surmounted  by  pinnacles,  rendering  them  thus  a  suffi- 
ciently steady  abutment  to  the  flying  arches,  which,  in  their  turn,  abutted 
the  central  vaults. 

An  easy  transition  leads  us  from  pinnacles  to  spires,  the  latter  being 
but  the  perfect  development  of  the  former,  and  each  requiring  the  assist- 
ance of  the  other  in  producing  a  thoroughly  harmonious  effect.  Yet  the 
spire  never  was  a  constructive  expedient,  or  useful  in  any  way.  From  the 
tower,  the  spire  arose  first  as  a  wooden  roof,  and  as  height  was  one  of  the 
great  objects  to  be  attained,  it  was  carried  to  an  elevation  beyond  the 
mere  requirements  of  a  protection  against  the  weather. 

The  earlier  towers  of  the  Romanesque  style  were  constructed  without 
spires.  All  are  square  in  plan,  and  extremely  similar  in  design.  Fig.  6, 


268  ARCHITECTURAL    DRAWING. 

Plate  LYIL,  is  an  elevation  of  the  tower  attached  to  the  church  of  Sta. 
Maria,  in  Cosmedin,  and  is  one  of  the  best  and  most  complete  examples 
of  this  style.  Its  dimensions  are  small,  being  but  15  feet  broad  and  110 
feet  high  ;  a  sufficiency  of  height,  where  buildings  are  not  generally  tall, 
to  give  prominence,  without  overpowering  other  objects,  which  renders 
these  towers  not  only  beautiful  structures  in  themselves,  but  singularly 
appropriate  ornaments  to  the  buildings  to  which  they  were  attached. 
Those  towers  are  the  types  of  the  later  Italian  campaniles,  or  bell-towers, 
most  generally  attached  to  some  angle  of  churches,  but  sometimes  de- 
tached, yet  so  placed  that  they  still  form  a  part  of  the  church  design. 
Sometimes  they  are  but  civic  constructions,  as  belfries,  or  towers  of  de- 
fence. In  design,  the  Gothic  towers  differ  from  the  Italian  campaniles. 
The  campanile  is  square,  carried  up  without  break  or  offset,  to  two-thirds, 
at  least,  of  its  intended  height ;  it  is  generally  solid  to  a  considerable  height, 
or  with  only  such  openings  as  serve  to  admit  light  to  the  staircases.  Above 
this  solid  part  one  round  window  is  introduced  in  each  face,  in  the  next 
story,  two,  in  the  one  above  this,  three,  then  four,  and  lastly,  five,  the 
lights  being  separated  by  slight  piers,  so  that  the  upper  story  is,  virtually, 
an  open  loggia. 

The  Gothic  towers  have  projecting  buttresses,  frequent  offsets,  lofty 
spires,  and  a  general  pyramidal  form.  Fig.  7  is  the  front  elevation  of  a 
simple  English  Gothic  tower ;  here  the  plain  pyramidal  roof,  rising  at 
an  equal  slope  on  each  of  the  four  sides,  is  intersected  by  an  octagonal 
spire  of  steep  pitch.  The  first  spires  were  simple  quadrangular  pyramids, 
afterwards  the  angles  were  cut  off,  and  they  became  octagonal,  and  this  is 
the  general  Gothic  form  of  spire.  Often  instead  of  intersecting  the  square 
roof  as  in  the  figure,  the  octagonal  spire  rests  upon  a  square  base,  and  the 
angles  of  the  tower  are  carried  up  by  pinnacles,  or  the  sides  by  battlements, 
or  by  both,  as  in  fig.  8,  to  soften  the  transition  between  the  perpendicular 
and  sloping  part. 

In  general  the  spires  of  English  churches  are  more  lofty  than  those  on 
the  Continent.  The  angle  at  the  apex  in  the  former  being  about  10°  and  in 
the  latter,  about  15°.  The  apex  angle  of  the  spires  of  Chichester  and  Lich- 
field,  are  from  12°  to  13°,  or  a  mean  between  the  two  proportions,  and  ac- 
cording to  Ferguson,  more  pleasing  than  either;  although  having  more  lofty 
spires,  yet  the  English  construction  is  much  more  massive  in  appearance, 
than  the  Continental ;  the  apertures  are  less  numerous,  and  the  surfaces  are 
less  cut  up,  and  covered  with  ornaments.  The  spires  of  Friberg  Church 
and  many  others  on  the  Continent  are  made  open  work,  a  precedent  fol- 
lowed sometimes  in  this  country,  but  not  in  the  same  material — wood 


ARCHITECTURAL    DRAWING.  269 

rather  than  stone.  In  cast  iron,  the  same  effect  would  be  obtained  at  a  less 
cost,  and  equally  durable  with  stone. 

Sometimes  the  central  spires  of  the  tower  were  omitted ;  each  of  the 
pinnacles  at  the  angles  being  converted  as  it  were  into  spires.  Sometimes 
the  tower  is  abruptly  ended  by  mere  battlements  around  its  sides.  In  the 
poorer  churches,  a  bell-cot  was  made  to  serve  the  purpose  of  a  bell  tower ; 
this  was  formed  by  carrying  up  the  gable  wall,  as  in  fig.  9,  and  making 
apertures  for  the  reception  of  the  bell.  When  the  wall  was  not  of  the  re- 
quisite thickness,  the  cot  was  either  supported  by  buttresses  from  beneath, 
or  the  corbels  were  projected  from  each  side  of  the  wall. 

Fig.  10  represents  the  upper  portion  of  the  tower  of  Ivan  Yeliki  at 
Moscow.  The  Russian  towers  are  generally  constructed  independent  of 
their  churches,  and  are  intended  for  the  reception  of  their  massive  bells. 

Windows. — Before  the  use  of  painted  glass,  very  small  apertures  suf- 
ficed for  the  introduction  of  the  required  quantity  of  light  into  a  church  ; 
as  a  consequence  the  windows  of  the  Romanesque  churches  were  gener- 
ally small,  and  devoid  of  tracery.  Again,  as  the  Byzantine  architects 
adorned  their  walls  with  paintings,  they  could  not  make  use  of  stained 
glass ;  neither  in  their  climate,  did  they  require  large  apertures ;  they  fol- 
lowed in  general  form  the  Romanesque  window,  apertures  with  circular 
heads,  either  single  or  in  groups  (fig.  1,  Plate  LVIII.  or  fig.  6,  Plate  LYII). 
The  Korman  windows  were  also  small,  each  consisting  of  a  single  light, 
semicircular  in  the  head,  and  placed  as  high  as  possible  above  the  ground  ; 
at  first  splayed  on  the  inside  only,  afterwards  the  windows  began  to  be 
recessed  with  mouldings  and  jamb  shafts  in  the  angles,  as  in  fig.  2. 

The  Lancet  in  general  use  in  the  early  Gothic  period  was  of  the  sim- 
plest arrangement :  in  these  windows  the  glass  was  brought  within  three 
or  four  inches  of  the  outside  of  the  wall,  and  the  openings  were  widely 
splayed  in  the  interior.  The  proportions  of  these  windows  vary  consider- 
ably ;  in  some  the  height  being  but  five  times  the  width,  in  others  as  much 
as  eleven  ;  eight  or  nine  times  may  be  taken  as  the  average.  Lancet  win- 
dows occur  singly :  in  groups  of  two,  three,  five  and  seven,  rarely  of  four 
and  six.  The  triplet,  fig.  3,  is  the  most  beautiful  arrangement  of  lancet 
windows.  It  was  customary  to  mark  with  greater  importance  the  central 
light,  by  giving  it  additional-  height,  and  in  most  cases  increased  width 
also.  In  some  examples  the  windows  of  a  lancet  triplet  are  placed  within 
one  dripstone  forming  a  single  arch,  thus  bearing  a  strong  resemblance  to 
a  single  three-light  window.  The  first  approximation  to  tracery  appears  to 
have  been  the  piercing  of  the  space  over  a  double  lancet  window  com- 
prised within  a  single  dripstone ;  in  place  of  the  customary  simple 


270  ARCHITECTURAL   DRAWING. 

arch  head,  in  some  examples  of  lancet  windows,  the  head  of  the  light  is 
foiled. 

From  the  combination  and  foiling,  or  cusping,  of  distinct  lancets,  a 
single  window  divided  by  mullions  and  tracery  derives  its  origin. 

A  traceried  window  may  be  justly  regarded  as  a  distinctive  character- 
istic of  Gothic  architecture.  With  the  decided  establishment  of  the  prin- 
ciple of  window  tracery,  it  became  a  recognized  constructive  arrangement 
to  recess  the  mullions  from  the  face  of  the  wall  in  which  the  window  arch 
was  pierced,  and  the  fine  effect  thus  produced  was,  as  the  art  advanced, 
speedily  enhanced  by  the  introduction  of  distinct  orders  of  mullions,  and 
by  recessing  certain  portions  of  the  tracery  from  the  face  of  the  primary 
mullions  and  their  corresponding  tracery  bars.  The  tracery  bars  are  those 
portions  of  the  masonry  of  the  window  head  which  mark  out  the  principal 
figures  of  the  design ;  from  these  the  minor  and  more  strictly  decorative 
parts  of  the  stone  work  may  be  distinguished  under  the  title  of  Form 
pieces. 

Decorated  window  tracery  has  been  generally  divided  into  two  chief 
varieties,  Geometrical  and  Flowing ;  the  former  consisting  of  geometrical 
figures,  as  circles,  trefoils,  quatrefoils,  curvilinear  triangles,  lozenges,  &c. 
&c. ;  while  in  flowing  tracery,  these  figures,  though  still  existing,  are 
gracefully  blended  together  in  one  design.  In  its  most  perfect  state,  geo- 
metrical tracery  invariably  exhibits  some  large  figure  of  a  distinct  and  de- 
cided character,  which  occupies  the  entire  upper  part  of  the  window-head. 

Fig.  4  represents  a  quatrefoil  window,  fig.  5,  a  pointed  trefoil  in  out- 
line ;  with  the  centres  of  the  different  circles  indicated,  and  such  lines  as 
may  be  necessary  to  explain  the  way  in  which  they  are  described.  These 
forms  and  modifications  of  them,  will  be  found  of  general  application  in 
traceried  windows.  Fig.  6  represents  two  forms  of  circular  windows,  or 
roses  tournantes. 

Fig.  7  represents  an  example  of  the  earlier  decorated  tracery  window- 
head,  consisting  of  two  foiled  lancets,  with  a  pointed  quatrefoil  in  the 
spandrel  between  them.  One  half  of  the  windows  in  this,  as  in  some  of 
the  following  figures,  is  drawn  in  skeleton  to  explain  their  construction. 

Fig.  8  is  another  example  of  Decorated  tracery. 

Fig.  9  is  an  example  of  the  English  leaf,  tracery ;  fig.  10  of  the  French 
flamboyant.  The  difference  between  the  two  styles  is,  that  while  the 
upper  ends  of  the  English  loops  or  leaves  are  round,  or  simply  pointed, 
the  upper  ends  of  the  latter  terminate  like  their  lower  ones,  in  angles  of 
contact,  giving  a  flame-like  form  to  the  tracery  bars  and  form  pieces. 

In  England  the  Perpendicular  style  succeeded  the  Decorated ;  the  mul- 


ARCHITECTURAL   DRAWING.  271 

lions  instead  of  diverging  in  flowing  or  curvilinear  lines,  are  carried  up 
straight  through  the  head  of  the  windows  ;  smaller  mullions  spring  from 
the  head  of  the  principal  lights,  and  thus  the  upper  portion  of  the  window 
is  filled  with  panel-like  compartments.  The  principal  as  well  as  the  sub- 
ordinate lights  are  foliated  in  their  heads,  and  large  windows  are  often  di- 
vided horizontally  by  transoms.  The  forms  of  the  window  arches  vary 
from  simple  pointed,  to  the  complex  four-centred,  more  or  less  depressed. 

Fig.  11  is  an  example  of  Perpendicular  windows. 

Fig.  12  is  a  square-headed  window,  such  as  were  usual  in  the  clere 
stories  of  Perpendicular  architecture. 

Figs.  13  and  14  are  quadrants  of  circular  windows,  used  more  especial- 
ly in  France,  for  the  adornment  of  the  west  ends  and  transepts  of  the  ca- 
thedrals. 

Besides  the  tracery  characteristic  of  Gothic  architecture,  there  is  a 
tracery  peculiar  to  the  Saracenic  and  Moorish  style,  of  which  fig.  15  may 
be  taken  as  an  example — it  being  a  window  of  one  of  the  earliest  mosques. 
The  general  form  of  the  window  and  door-heads  of  this  style  is  that  of  the 
horse-shoe,  either  circular  or  pointed. 

Doorways. — Plate  LIX.  fig.  1,  is  the  elevation  of  a  circular-headed 
doorway,  which  may  be  considered  the  type  of  many  entrances  both  in 
Romanesque,  Gothic,  and  later  styles.  It  consists  of  two  or  more  recessed 
arches,  with  shafts  or  mouldings  in  the  jambs.  In  the  earlier  styles  the 
arches  were  circular,  in  the  later  Gothic,  generally  pointed,  but  some- 
times circular ;  in  the  earlier,  the  angles  in  which  the  shafts  are  placed  are 
rectangular  ;  in  the  later,  the  shaft  is  often  moulded  on  a  chamfer  plane, 
that  is,  a  plane  inclined  to  the  face  of  the  wall,  generally  at  an  angle  of 
45°  ;  often  the  chamfer  and  rectangular  planes  are  used  in  connection. 

Fig.  2  is  a  simple  head  of  a  depressed  four-centred  or  Tudor-arched 
doorway,  with  a  hood  moulding.  Fig.  3  represents  the  incorporation  of 
a  window  and  doorway.  Sometimes  the  doorway  pierces  a  buttress ;  in 
that  case,  the  buttress  expands  on  either  side  forming  a  sort  of  porch.  The 
Gothic  architects  placed  doors  where  they  were  necessary,  and  made  them 
subservient  to  the  beauty  of  the  design. 

Fig.  4  is  an  example  of  a  gabled  doorway  with  crockets  and  finials. 
Fig.  5,  of  a  Perpendicular  doorway,  with  a  label  or  hood  moulding  above, 
and  ornamented  spandrels. 

Fig.  6  is  an  example  of  a  Byzantine,  and  fig.  7  of  a  Saracenic  doorway. 

The  Renaissance  style  succeeded  the  Gothic,  being,  originally,  but  the 
revival  or  a  fair  rendering  of  the  classical  orders  of  architecture,  with  or- 
naments from  the  Byzantine  and  Saracenic  styles. 


272  ARCHITECT-HEAL   DRAWING. 

It  was  in  Italy  tliat  this  revival  took  place,  and  Garbett  divides  this 
style  into  three  Italian  schools,  the  Florentine,  Venetian,  and  Roman,  ex- 
hibiting a  certain  analogy  to  the  three  orders  of  ancient  architecture. 
The  Florentine,  corresponding  to  the  Doric,  admits  of  little  apparent 
ornament,  but  any  degree  of  real  richness,  preserving  in  its  principal 
forms  severe  contrast;  powerful  masses  self-poised  without  corbelling, 
without  arching ;  breadth  of  every  thing,  of  light,  of  shade,  of  ornament, 
of  plain  wall ;  depth  of  recess  in  the  openings,  of  perspective  in  the  whole 
mass,  of  projection  in  the  cornice.  To  these  add  a  sort  of  utilitarianism, 
or  absence  of  features  useless  to  convenience  or  stability,  an  absence  of 
sacrifice  of  material,  admitting  of  great  plainness,  of  very  florid  enrich- 
ment. On  the  whole,  the  Florentine  may  be  called  the  plain,  common- 
sense  school. 

Very  different  in  principle  was  the  Venetian  school,  which,  like  its 
prototype,  the  Corinthian,  superseded  its  sober  rivals.  Its  aim  was  splen- 
dor, variety,  show,  and  ornament ;  not  so  much  real  as  effective  ornament. 
Thus,  it  rarely  contains  so  much  carving  or  minute  enrichment  as  the  Flo- 
rentine admits ;  but  it  has  larger  ornaments,  constructed  (or  built)  orna- 
ments, great  features  useless  except  for  ornament,  as  inaccessible  porticoes, 
detached  columns,  and  architraves  supporting  no  ceiling,  towers  built  only 
for  breaking  an  outline.  Its  decoration  is  spread  equally  over  the  whole 
work.  Rectangular  severity  gives  place  to  curved  elegance,  in  arches, 
domes,  circular  and  oblique-angled  plans,  true  grandeur  to  effect,  intellec- 
tual sense  of  fitness  to  eumorphic  beauty. 

The  Roman  school,  holding  the  same  place  as  the  Ionic,  is  intermediate 
in  every  respect  between  the  two  other  schools.  It  is  better  adapted  to 
churches  than  to  any  other  class  of  buildings.  This  fitness  arises  from  the 
grand,  simple,  and  unitory  effect  of  one  tall  order,  generally  commencing 
at  or  near  the  ground,  and  including,  or  rather  obliterating,  the  distinction 
'of  two  or  three  stories,  making  a  high  building  appear  a  single  story. 

To  describe  these  schools  technically,  the  Florentine  is  mostly  astylar, 
the  style  of  finistration  and  rustic  quoins ;  the  Roman,  the  style  of  pilas- 
ters ;  and  the  Venetian,  that  of  columns.  In  calling  the  Florentine  asty- 
lar, a  total  absence  of  external  orders  is  not  implied,  but  their  absence  as 
main  features,  or  on  a  considerable  scale.  Their  chief  application  is  to 
windows  and  doors,  and  the  greatest  orders  never  include  so  much  as  the 
height  of  a  single  story.  In  the  Roman  school,  the  great  scale  of  the 
principal  order  renders  it  chiefly  an  order  of  pilasters.  The  outer  pilasters 
of  the  great  order  were  often  filled  in  with  smaller  and  columnar  ones,  in 
two  tiers,  while  a  still  smaller  set  decorated  the  openings.  As  the  Vene- 


ARCHITECTURAL   DRAWING.  273 

tians  did  not  use  such  large  orders,  they  easily  made  them  more  columnar, 
and  introduced  hanging  entablatures.  In  this  school  there  is  (except  in 
churches)  no  principal  story  or  order,  if  there  be  more  than  one,  all  are 
nearly  equal,  or  equally  important. 

General  plan  and  outline,  in  the  Florentine,  is  of  the  utmost  simplicity, 
rendering  it  fitter  for  town  than  country  buildings ;  in  the  Koman,  slightly 
more  varied ;  in  the  Venetian,  whenever  the  site  will  admit,  broken, 
complex,  and  picturesque. 

We  have  thus  briefly  treated  of  the  distinguishing  features,  according 
to  Garbett,  of  the  three  modern  schools ;  there  are  many  other  distinctive 
styles  and  names,  but  they  may  mostly  be  included  under  the  one  or  the 
other  of  these  schools,  their  claim  for  a  distinctive  name  resting  rather  on 
the  peculiar  style  of  ornament  or  tracery  used,  than  any  great  distinctive 
architectural  feature. 

Ornament. — Architectural  ornament  is  of  two  kinds,  constructive  and 
decorative.  By  the  former  is  meant  all  those  contrivances,  such  as  capi- 
tals, brackets,  vaulting  shafts,  and  the  like,  which  serve  to  explain  or  give 
expression  to  the  construction ;  by  the  latter,  such  as  mouldings,  frets, 
foliage,  &c.,  which  give  grace  and  life,  either  to  this  actual  constructive 
forms,  or  to  the  constructive  decoration.  It  is  to  the  latter  class  that  we 
wish  to  call  attention ;  mouldings  of  the  different  styles  have  been  already 
treated  of;  we  therefore  propose  to  give  now  what  are  even  more  merely 
decorations  of  a  style. 

First,  as  to  Grecian  orders.  By  reference  to  Plate  LIII.  we  see  that 
the  Doric  has  the  triglyph  mutules  and  guttse.  By  reference  to  Plate 
LIV.,  the  Ionic,  we  find  various  mouldings  of  the  cornice  frieze,  abacus, 
and  neck  of  the  column  enriched.  The  principal  ornament  of  the  neck 
of  the  column  is  the  anthemion,  commonly  known,  in  its  most  simple 
form,  as  the  honeysuckle  or  palmetto  ;  in  the  anthemion  as  represented  in 
the  figure,  the  palmetto  alternates  with  the  lily  or  some  analogous  form. 
The  ornament  of  the  abacus  is  the  egg  and  dart,  shown  on  a  large  scale, 
fig.  9,  Plate  LX.,  where  may  be  found  also  the 
ornament  of  the  frieze  and  cornice,  fig.  7.  Fig.  99, 
the  fret,  and  fig.  100,  the  guilloche,  are  also  com- 


mon    Greek  ornaments,  used  to  adorn  the   soffits  ,Fis-  "• 

of  beams,  and  ceilings.  The  acanthus  is  the  dis- 
tinctive ornament  of  the  Corinthian,  of  which  a 
leaf  is  represented  on  a  large  scale  in  front  and 
side  view,  figs.  1,  2,  3,  Plate  LX.  These  figures 

illustrate,   also,   the  way  in  which   ornaments  of 
18 


274:  ARCHITECTURAL   DRAWING. 

irregular  figure  are  copied  by  the  draughtsman.  Thus,  suppose  it  were 
required  to  draw  fig.  2,  and  in  a  reversed  position ;  circumscribe  around 
the  given  figure,  a  parallelogram ;  divide  this  parallelogram  into  any 
number  of  equal  squares,  and  in  the  position  required  for  the  copy,  as 
fig.  3  for  instance,  construct  a  similar  parallelogram.  For  convenience 
of  reference,  we  have  marked  the  vertical  divisions  of  the  squares  by  let- 
ters, a,  5,  c,  d,  e,f,  g,  and  the  horizontal  divisions  by  figures,  1,  2,  3,  4 ;  this 
is  done  with  both  parallelograms.  But  it  must  be  remarked,  that  if  the 
copy  is  to  be  a  reverse  of  the  original,  the  figures  marking  the  horizontal 
divisions  of  the  copy  must  be  the  reverse  of  the  original,  as  may  be  seen 
in  figs.  2  and  3.  Now  mark  on  the  different  vertical  and  horizontal  lines 
of  the  corresponding  squares,  the  relative  positions  of  the  parts  of  the 
leaf,  and  through  these  points  thus  established,  construct  the  leaf  required. 
A  similar  method  may  be  used  in  constructing  a  copy  to  an  enlarged 
or  to  a  reduced  size  of  the  original,  by  enlarging  or  reducing  the  compar- 
ative sizes  of  the  squares  of  the  parallelogram  of  the  copy  on  a  scale  pro- 
portioned to  the  enlargement  or  reduction  required.  In  general,  the  inter- 
sections of  the  portions  of  the  leaf,  or  other  figure,  with  the  vertical  or 
horizontal  lines,  are  measured  and  transferred  by  the  eye ;  the  larger  the 
number  of  squares,  therefore,  the  greater  probability  of  the  copy  coinciding 
with  the  original.  Figs.  4,  5,  and  6,  are  the  side  elevation,  front  elevation, 
and  section  of  a  Greek  bracket,  the  principal  ornaments  of  which  are  taken 
from  the  anthemion  and  acanthus. 

Fig.  1,  Plate  LXL,  is  an  elevation  of  a  portion  of  an  enriched  cornice 
from  the  temple  of  Jupiter  Stator  at  Rome,  of  the  Corinthian  order  of  ar- 
chitecture. Fig.  2,  is  the  under  side  of  the  modillion. 

The  chief  characteristic  of  Roman  ornament,  is  its  uniform  magnifi- 
cence. As  a  style  it  is  not  original,  but  rather  an  enlargement  or  enrich- 
ment of  the  Greek.  There  is,  further,  this  distinction  between  the  two 
styles,  that  the  most  rarely  used  elements  among  the  Greeks  are  the  most 
characteristic  of  the  Roman  decorations,  the  scroll  and  the  acanthus.  In- 
deed, every  form  which  will  admit  of  it,  is  habitually  enriched  with  acan- 
thus clothing  or  foliations.  The  acanthus  of  the  Greeks  is  the  narrow 
prickly  acanthus ;  that  of  the  Roman,  the  soft  acanthus.  For  capitals  the 
Roman  acanthus  is  commonly  composed  of  conventional  clusters  of  olive 
leaves.  The  Greek  scroll  is  seldom  elaborated,  but  the  Roman  is  seldom 
without  acanthus  foliations.  Fig.  3,  represents  a  Roman  acanthus  scroll. 

The  free  introduction  of  monsters  and  animals  is  likewise  a  character- 
istic of  Greek  and  Roman  ornament,  as  the  sphinx,  the  triton,  the  griffin, 
and  others ;  they  occur  however  more  abundantly  in  the  Roman. 


ARCHITECTURAL   DRAWING.  275 

As  the  Christian  art  succeeded  the  Pagan,  symbols  became  the  founda- 
tion of  decorations  in  the  Byzantine  and  Romanesque.  The  early  sym- 
bols were  the  monogram  of  Christ,  the  lily,  the  cross,  the  serpent,  the  fish, 
the  aureole,  or  vesica  piscis,  and  the  circle  or  nimbus,  the  glory  of 
the  head,  as  the  vesica  is  of  the  whole  body.  These  are  very  important 
'elements  in  Christian  decoration,  especially  the  nimbus,  which  is  the  ele- 
ment of  the  trefoil  and  quatrefoil ;  the  first  having  reference  to  the  Trinity, 
the  Jfecond  to  the  four  Evangelists,  as  the  testimony  of  Christ,  and  to  the 
Cross,  at  the  extremities  of  which  we  often  find  four  circles,  besides  the 
circle  in  the  centre,  which  signifies  the  Lord.  Occasionally  the  symbolic 
images  of  the  Evangelists,  the  angel,  the  lion,  the  ox,  and  the  eagle,  are 
represented  within  these  circles. 

The  hand  in  the  attitude  of  benediction,  and  the  lily  (the  fleur-de-lis), 
the  emblem  of  the  virgin  and  purity,  are  common  in  Christian  decoration. 
This  last  symbol  was  eventually  elaborated  into  the  most  characteristic 
foliage  of  Byzantine  and  Romanesque  art.  Conspicuous  in  their  foliage 
also  is  a  peculiar  formed  leaf,  somewhat  resembling  the  leaf  of  the  ordinary 
thistle.  The  serpent  figures  largely  in  Byzantine  art  as  the  instrument  of 
the  fall,  and  one  type  of  the  redemption. 

As  paganism  disappeared,  their  ornaments,  under  certain  symbolic 
modifications,  were  admitted  into  Christian  decorations.  Thus  the  folia- 
tions of  the  scroll  were  terminated  by  lilies,  or  by  leaves  of  three,  four  and 
five  blades,  the  number  of  blades  being  significant ;  and  in  a  similar  way, 
the  anthemion  and  every  other  ancient  ornament.  In  the  Byzantine  the 
symbolism  is  seldom  or  ever  absent,  however  much  it  may  be  modified  or 
disguised.  An  important  feature,  always  to  be  observed  in  the  Byzantine, 
is  that  all  their  imitations  of  natural  forms  were  invariably  conventional ; 
it  is  the  same  even  with  animals  and  the  human  figure,  every  saint  had  his 
prescribed  colors,  proportions  and  symbols. 

The  Saracenic  was  the  period  of  gorgeous  diapers,  for  their  habit  of 
decorating  the  entire  surfaces  of  their  apartments  was  highly  favorable  to 
the  development  of  this  class  of  design.  The  Alhambra  displays  almost 
endless  specimens,  and  all  are  in  relief  and  enriched  with  gold  and  color, 
chiefly  blue  and  red.  The  religious  cycles  and  symbolic  figures  of  the  By- 
zantine are  excluded.  Mere  curves  and  angles  or  interlacings  were  now 
to  bear  the  chief  burden  of  a  design,  but  distinguished  by  a  variety  of  co- 
lor. The  curves  however  very  naturally  fell  into  standard  forms  and  floral 
shapes,  and  the  lines  and  angles  were  soon  developed  into  a  very  charac- 
teristic species  of  tracery,  or  interlaid  strap  work,  very  agreeably  diversified 
by  the  ornamental  introduction  of  the  inscriptions,  which  last  custom  of 


ARCHITECTURAL   DRAWING. 


elaborating  inscriptions  with  their  designs  was  peculiarly  Saracenic.  Al- 
though flowers  were  not  palpably  admitted,  yet  the  great  mass  of  the 
minor  details  of  Saracenic  designs  are  composed  of  flower  forms  disguised, 
the  very  inscriptions  are  sometimes  thus  grouped  as  flowers ;  still  no  ac- 
tual flo'wer  ever  occurs,  as  the  exclusion  of  all  natural  images  is  funda- 
mental to  the  style  in  its  purity. 

Fig.  3,  is  a  specimen  of  Alhambra  diaper. 

All  the  symbolic  elements  of  the  Byzantine  are  continued  in  the  Gallic. 
Ornamentally,  the  Gothic  is  the  geometrical  and  pointed  element  elaborated 
to  the  utmost ;  its  only  peculiarities  are  its  combinations  of  details ;  at  first 
the  conventional  and  geometrical  prevailing,  and  afterwards  these  com- 
bined with  the  elaboration  of  natural  objects  in  its  decoration.  The  By- 
zantines never  did  this,  their  ornaments  are  purely  conventional ;  while 
in  the  finest  gothic  specimens,  not  only  the  traditional  conventional  orna- 
ments, but  also  elaborate  imitations  of  natural  plants  and  flowers  are  found. 
The  most  striking  feature  of  all  Gothic  work  is  the  wonderful  elaboration 
of  its  geometric  tracery ;  vesicas,  trefoils,  quatrefoils,  cinquefoils,  and  an 
infinity  of  geometric  varieties  besides.  The  tracery  is  so  paramount  a 
characteristic,  that  the  three  English  varieties,  the  early  English,  the  deco- 
rated, and  the  perpendicular,  and  the  French  flamboyant,  are  distinguished 
almost  exclusively  by  this  feature.  See  Plate  LYIII. 

Under  the  head  of  Gothic,  the  Norman  is  often  included,  but  it  is  rather 
a  transition  style  between  the  Romanesque  or  Byzantine,  and  the  Gothic. 
The  ornamental  mouldings  used  in  the  decorative  details  of  this  style  are 
numerous,  among  which  the  more  common  is  the  chevron  or  zig-zag,  (fig. 
1,  plate  LXII.,)  simple  as  the  indented,  or  duplicated,  triplicated  or  quad- 
rupled ;  the  billet,  the  prismatic  billet,  the  square  billet,  and  the  alternate 
billet  (fig.  2) ;  the  star  fig.  3,  the  fir  cone ;  the  cable,  fig.  4 ;  the  embat- 
tled, fig.  5 ;  the  nail  head,  fig.  6.  In  the  early  English  style  we  find  the 
dog-tooth,  fig  T ;  a  kind  of  pyramid-shaped  flower  leaves ;  the  ball  flower, 
fig.  8,  and  the  serpentine  vine  scroll,  are  the  most  characteristic  ornamen- 
tal mouldings  of  the  decorated  style.  The  mouldings  of  the  perpendicu- 
lar are  not  peculiar ;  they  are  less  enriched  than  the  preceding  styles,  and 
the  same  panelling  which  is  found  in  the  windows  is  spread  over  every 
surface  of  the  building. 

In  the  early  English  we  have  the  first  development  of  geometrical  tracery, 
flying  buttresses,  crocketed  pinnacles,  columns  clustered,  and  an  extensive 
application  of  foliage  with  the  trefoil  leaf,  as  the  most  characteristic  orna- 
ment ;  sometimes  formed  as  a  clover  leaf,  at  other  times  very  irregularly 
formed. 


ARCHITECTURAL   DRAWING.  277 

The  early  English  is  characterized,  besides  its  tracery,  by  the  ogee  and 
the  pinnacled  canopied  recesses  of  its  buttresses  and  other  parts  producing 
a  prominence  of  diagonal  lines.  There  is  also  more  copying  of  nature  in 
its  ornamental  details. 

In  the  Perpendicular,  the  new  features  are  the  horizontal  line  and  the 
panellings,  and  the  substitution  of  perpendicular  for  flowing  tracery. 

The  crocket,  in  its  earliest  form,  was  the  simple  arrow  head  of  the 
Episcopal,  pastoral  staff;  subsequently  finished  with  a  trefoil,  and  after- 
wards still  further  enriched.  Figs.  9  and  10  are  early  English  crockets  ; 
fig.  11  a  decorated  one.  Fig.  12  is  a  finial  of  the  same  style ;  both  finials 
and  crockets  in  detail  display  a  variety  of  forms ;  some  resembling  the 
botanical  productions  of  one  class,  some  of  another. 

The  Parapets  of  the  early  English  style  are  often  a  simple  horizontal 
course,  supported  by  a  corbel  table,  sometimes  relieved  by  a  series  of  sunk 
blank  trefoil-headed  panels ;  sometimes  a  low  embattled  parapet  crowns 
the  wall.  In  the  decorated  style,  the  horizontal  parapet  is  sometimes 
pierced  with  trefoils,  sometimes  with  wavy  flowing  tracery  (fig.  13).  Gro- 
tesque spouts  or  gargoyles  discharge  the  water  from  the  gutters.  The 
parapets  of  the  perpendicular  style  are  frequently  embattled  (14),  covered 
with  sunk  or  pierced  panelling,  and  ornamented  with  quatrefoil,  or  small 
trefoil-headed  arches ;  sometimes  not  embattled  but  covered  with  sunk  or 
pierced  quatrefoils  in  circles,  or  with  trefoils  in  triangular  spaces  as  in 
fig.  15. 

Amongst  the  varieties  of  ornamental  work,  the  mode  of  covering  small 
plain  surfaces  with  diapering  (fig.  16),  was  sometimes  used ;  the  design 
being  in  exact  accordance  with  the  architectural  features  and  details  of 
the  style.  The  rose,  fig.  17,  the  badge  of  the  houses  of  York  and  Lancas- 
ter, is  often  met  with  in  the  perpendicular  style  ;  and  tendrils,  leaves  and 
fruit  of  the  vine,  are  carved  in  great  profusion  in  the  hollows  of  rich  cor- 
nice mouldings,  especially  on  screen  work  in  the  interior  of  a  church. 
Fig.  18,  in  its  original  type,  a  Byzantine  ornament,  an  alternate  lily  and 
cross,  is  a  common  finish  to  the  cornice  of  rich  screen  work  in  the  latest 
Gothic,  and  is  known  under  the  name  of  the  Tudor  flower. 

Figs.  19,  20,  21,  are  examples  of  ornamental  crosses  used  as  finials, 
either  for  spires  or  pinnacles. 

The  Ornaments  of  the  Renaissance. — The  term  Renaissance  is  used  in  a 
double  sense ;  in  a  general  sense  implying  the  revival  of  art,  and  specially, 
signifying  a  peculiar  style  of  ornament.  It  is  also  sometimes,  in  a  veiy 
confined  sense,  applied  in  reference  to  ornament  to  the  style  of  Benvenuto 
Cellini ;  or,  as  it  is  sometimes  designated,  the  Henry  II.  (of  France)  style. 


ARCHITECTURAL    DRAWING. 


The  mixture  of  various  elements  is  one  of  the  essentials  of  this  style. 
These  elements  are  the  classical  ornaments  ;  unnatural  and  natural  flowers 
and  foliage,  the  former  often  of  a  pure  Saracenic  character;  man  and  ani- 
mals, natural  and  grotesque ;  cartouches,  or  pierced  and  scrolled  shields,  in 
great  prominence ;  tracery  independent,  and  developed  from  the  scrolls  of 
the  cartouches  ;  and  jewel  forms.  Fig.  1,  and  3,  Plate  LXIII. 

The  Elizabethan  is  a  partial  elaboration  of  the  same  style,  the  only 
difference  being  that  what  we  now  term  the  Elizabethan  exhibits  a  very 
striking  preponderance  of  strap  and  shield  work,  but  the  earlier  and  pure 
Elizabethan  is  much  nearer  allied  to  the  continental  styles  of  the  time ; 
classical  ornaments  but  rude  in  detail,  occasional  scroll  and  arabesque 
work,  and  strap  work,  holding  a  much  more  prominent  place  than  the 
pierced  or  scrolled  shields.  Fig.  2  is  an  example  of  the  style  from  the  old 
guard  chamber,  Westminster. 

Of  the  earliest  and  transition  styles  of  Renaissance  ornament,  are  the 
Tricento  and  the  Quatrecento  ;  the  great  features  of  the  first  are  its  intricate 
tracery  and  delicate  scroll  work  of  conventional  foliage,  the  style  being 
but  a  slight  remove  from  the  Byzantine  and  Saracenic.  Of  the  second 
are,  in  addition,  elaborate  natural  imitations  of  fruit,  flowers,  birds  or 
animals  (fig.  4),  all  disposed  simply  with  a  view  to  the  ornamental ;  also 
occasional  cartouches,  or  scrolled  shield  work. 

In  all  these  styles,  the  evidence  of  their  Byzantine  or  Saracenic  origin 
is  constantly  preserved,  in  the  tracery,  in  the  scroll  work  and  foliage,  and 
in  the  rendering  of  classical  ornaments.  The  Eenaissance  is,  therefore, 
something  more  approximative  to  a  combination  of  previous  styles  than  a 
revival  of  .any  in  particular.  Yet  it  is  a  style  that  was  developed  solely 
on  aesthetic  principles,  from  a  love  of  the  forms  and  harmonies  themselves, 
as  varieties  of  effect  and  arrangements  of  beauty,  not  because  they  had 
any  particular  signification,  or  from  any  superstitious  attachment  to  them 
as  heirlooms. 

Fig.  5  is  an  example  of  ornament  in  the  Cinquecento  style.  The  ara- 
besque scroll  work  is  the  most  prominent  feature  of  the  Cinquecento,  and 
with  this  in  its  elements,  it  combines  every  other  feature  of  classical  art, 
with  the  unlimited  choice  of  natural  and  conventional  imitations  from 
the  entire  animal  and  vegetable  kingdom,  both  arbitrarily  disposed  and 
combined.  Absolute  works  of  art,  such  as  vases  and  implements,  and  in- 
struments of  all  kinds,  are  prominent  elements  of  the  Cinquecento  ara- 
besque, but  cartouches  and  strap  work  wholly  disappear  from  the  best 
examples.  Another  chief  feature  of  the  Cinquecento  is  the  admirable 
play  of  color  in  its  arabesques  and  scrolls,  and  it  is  worthy  of  note  that 


ARCHITECTURAL    DRAWING.  2  79 

the  three  secondary  colors,  orange,  green,  and  purple,  perform  the  chief 
parts  in  all  the  colored  decorations. 

Fig.  6  is  an  example  of  the  Louis  Quatorze  style  of  ornament.  The 
great  medium  of  this  style  was  gilt  stucco  work,  and  this  absence  of  color 
seems  to  have  led  to  its  most  striking  characteristic,  infinite  play  of  light, 
of  shade ;  color,  or  mere  beauty  of  form  in  detail,  having  no  part  in  it 
whatever.  Flat  surfaces  are  not  admitted ;  all  are  concave  or  convex :  this 
constant  varying  of  the  surface  gives  every  point  of  view  its  high  lights 
and  brilliant  contrasts. 

The  Louis  Quinze  style  differs  from  that  of  Louis  Quatorze  chiefly  in 
its  absence  of  symmetry ;  in  many  of  its  examples  it  is  an  almost  random 
dispersion  of  the  scroll  and  shell,  mixed  only  with  that  peculiar  crimping 
of  shell  work,  the  coquillage. 

The  ornaments  of  which  we  have  thus  given  examples  are,  in  general, 
applied  to  interior  decorations,  to  friezes,  pilasters,  panels,  architraves,  the 
faces  and  soffits  of  arches,  ceilings,  &c.,  to  furniture  and  to  art  manufac- 
tures in  general.  For  exteriors  these  ornaments  are  sparingly  applied  ; 
shield  and  scroll  work,  of  the  later  Elizabethan  or  Kenaissance  style,  is 
sometimes  used,  but  very  seldom  tracery. 

Of  common  exterior  ornament,  the  baluster  is  peculiarly  modern,  with 
all  the  refinement  of  a  classic  model.  Balustrades  are  sometimes  of  real 
use  in  building,  and  at  other  times  merely  ornamental.  Such  as  are  in- 
tended for  use,  as  when  they  are  employed  on  steps  or  stairs,  before  win- 
doM's,  or  to  enclose  terraces  or  other  elevated  places  of  resort,  must  always 
be  nearly  of  the  same  height,  from  three  to  three  and  a  half  feet,  so  that 
a  person  of  ordinary  height  may,  with  ease,  lean  over  them  without  the 
danger  of  falling.  But  those  that  are  principally  designed  for  ornament, 
as  when  they  finish  a  building ;  or  even  for  use  and  ornament,  as  when 
they  form  the  railing  over  a  large  bridge,  should  be  proportioned  to  the 
architecture  they  accompany,  and  their  height  ought  never  to  exceed  four- 
fifths  of  the  entablature  on  which  they  are  placed ;  nor  should  it  be  less 
than  two-thirds,  without  counting  the  plinth,  the  height  of  which  must  be 
sufficient  to  leave  the  whole  balustrade  exposed  to  view. 

Figs.  101,  102,  103,  104,  105,  and  106, 
represent  various  figures  of  balusters  and  of  va- 
rious proportions,  suited  to  the  various  orders 
they  may  serve  to  finish.  The  double-bellied 
balusters  (figs.  101  and  102)  are  the  lightest, 
and,  therefore  the  best  adapted  to  windows  Fig'101-  Fig'm  F1&m  Fis'104- 
or  other  compositions  of  which  the  parts  are  small  and  the  profiles 


280  AKCHITECTUEAL    DRAWING. 

delicate.     The  base  and  rail  may  be  of  the  same  profile  but  not  so  large 
as  for  single-bellied  ones. 

In  balustrades,  the  distance  between  two  balus- 
ters should  not  exceed  the  half  of  the  diameter  of 
the  thickest  part  of  the  baluster,  nor  less  than  one- 
third  of  it.  The  pedestals,  if  possible,  should  be  at 
intervals  of  about  nine  balusters,  but  as  the  pedes- 
tals must  be  placed  over  the  centre  of  the  piers,  the 
intervals  must  frequently  contain  more  balusters. 
Fig.  105.  Fig.  106.  -pig.  105  shows  the  arrangement  of  a  baluster 

with  inclined  rails  and  bases. 

"When  used  in  interiors,  either  for  decoration  or  use,  the  forms  of  the 
baluster  are  much  varied  and  enriched ;  this  is  especially  observable  in 
constructions  in  iron. 

ELEVATIONS     OP     HOUSES. 

Having  thus  given  a  brief  abstract  of  the  characteristics  of  various 
prominent  styles  of  architecture,  we  continue  our  article  on  houses  by 
giving  elevations,  either  suited  to  plans  already  exhibited,  or  to  other 
plans  which  will  be  found  on  the  same  plate  as  the  elevations.  It  must  be 
perceived  that,  in  general,  in  modern  constructions,  pure  ancient  art  is 
seldom  exhibited,  nor  would  it,  in  domestic  architecture,  be  found  suitable, 
the  requirements  and  appliances  being  very  different,  and  he  may  be 
called  an  architect,  who,  conversant  with  ancient  and  modern  practice, 
can  adapt  them  in  unity  and  harmony  to  modern  necessities. 

PL  LXIY.  represents  the  front  elevation  of  a  basement  house  with  the 
general  characteristics  of  the  Florentine  style,  uniting  richness  and  gran- 
deur of  effect,  admirably  suited  to  the  locality  and  purpose  for  which  it 
is  designed,  a  first-class  house,  or  even  what  might  be  termed  a  palatial 
residence.  This  building  has  been  constructed  in  Fifth  Avenue,  New 
York,  after  designs  of  T.  Thomas  &  Son,  architects ;  the  specifications  of 
which  will  be  found  in  a  subsequent  chapter. 

English  basement  houses  are  generally  constructed  with  a  rusticated 
basement  as  in  the  preceding  example,  (PL  LXIV.)  with  a  balustrade 
marking  the  distinction  between  it  and  the  principal  story.  The  entrance 
is  generally  raised  not  to  exceed  three  steps,  and  seldom  with  a  projecting 
porch ;  the  intention  being  to  make  the  basement  subordinate  to  the  prin- 
cipal story,  the  usual  finish  of  the  door-head  is  similar  to  that  of  the  win- 
dow. In  general  English  basement  houses  are  intended  for  narrow  lots ; 


AECHITECTUEAL   DRAWING.  281 

showing  in  front  but  two  windows  to  the  stories  above  the  basement,  and 
one  basement  window.  Circular  heads  are  almost  invariably  used  for  the 
basement  openings,  and  for  the  windows  above  either  square  or  slightly 
arched  lintels.  Sometimes  a  species  of  Romanesque  window  of  clustered 
openings  is  adopted. 

"When  the  architect  is  not  controlled  by  the  form  or  size  of  the  lot, 
much  picturesqueness  may  be  given  by  varieties  of  form  and  irregularities 
of  outline  in  the  construction  of  edifices. 

Plate  LXY.  is  an  elevation  of  a  house  from  Holly's  "  Country  Seats," 
after  a  style  of  architecture  usually  designated  here  as  the  French,  from 
the  form  of  the  Mansard  roof  and  its  dormer  windows,  rather  than  any  dis- 
tinctive features  in  the  main  elevation  or  its  ornaments.  This  style  of  roof 
is  very  effective,  and  has  become  very  popular ;  it  is  well  adapted  both  for 
city  and  country  residences. 

Plates  LXYL,  LXYIL,  LXYIII.,  and  LXIX.  are  elevations  and  plans  of 
country  residences  from  "  Downing's  Country  Houses,"  drawn  in  perspec- 
tive, the  principles  of  which  will  be  given  in  a  subsequent  chapter.  They 
may  be  taken  as  beautiful  illustrations  of  modern  constructions. 

Plate  LXYI.  are  the  plans  and  elevation  of  a  Farm  House  in  the  Eng- 
lish Rural  Style. 

Plate  LXYIL  is  an  elevation  and  plan  of  a  plain  timber  cottage  villa, 
after  designs  of  Gervase  Wheeler,  Architect,  of  Philadelphia.  "The 
construction,  though  simple,  is  somewhat  peculiar.  It  is  framed  in  such  a 
manner  that  on  the  exterior  the  construction  shows.  At  the  corners  are 
heavy  posts,  roughly  dressed  and  chamfered,  and  into  them  are  morticed 
horizontal  ties,  immediately  under  the  springing  of  the  roof ;  these,  with 
the  posts  and  the  studs  and  the  framing  of  the  roof  showing  externally. 
Internally  are  nailed  horizontal  braces  at  equal  distances  apart,  stopping 
on  the  posts  and  studs  of  the  frame,  and  across  these  the  furring  and  lath- 
ing cross  diagonally  in  different  directions.  On  these  horizontal  braces, 
the  sheathing  composed  of  plank  placed  in  a  perpendicular  position  is  sup- 
ported and  retained  in  its  place  by  battens,  two  and  a  half  inches  thick, 
and  made  with  a  broad  shoulder.  These  battens  are  pinned  to  the  hori- 
zontal braces,  confining  the  planks,  but  leaving  spaces  for  shrinking  and 
swelling,  thus  preventing  the  necessity  of  a  single  nail  being  driven 
through  the  planks.  A  representation  is  given  (fig.  107)  of  the  batten  B, 
and  the  mode  of  framing. 

Fig.  108  represents  the  usual  form  of  vertical  boarding,  which  is  less 
expensive  than  the  first  illustration,  and,  in  general,  will  be  found  suffi- 
ciently secured  for  the  class  of  buildings  to  which  it  is  applied. 


282  ARCHITECTURAL   DRAWING. 

Plate  LX YIII.  is  a  villa  in  what  Mr.  Downing  designates  as  the  Rural 


b 


Gothic  style,  designed  by  him- 
self. Figs.  109,  110,  111,  and 
112,  represent  some  of  the  de- 
tails on  a  larger  scale. 

Fig.  109  is  an  elevation  of 
the  bay  window  with  a  balco- 
ny over  it,  to  the  scale  of  one- 
quarter  of  an  inch  to  the  foot. 
Fig.  110,  the  verge  board  of 
the  small  gable  over  this  bal- 
cony. Fig.  Ill,  part  of  the 
verge  board  of  the  gable  over 
the  porch.  Tig.  112  are  chim- 
ney tops,  such  as  can  be  ob- 
tained of  Garnkirk  clay. 


Fig. 


Fig.  111. 


ABCHITECTUKAL   DEAWING. 


283 


w 

wW 

?m 

ra3 

C4$ 

i 


Fig.  112. 


a    a    B    Q 


Fig.  113. 


284 


ARCHITECTURAL   DRAWING. 


Plate  LXIX.  is  the  elevation  and  principal  floor  plan  of  a  villa  in  the 
Italian  style,  as  constructed  after  plans  of  Mr.  Upjohn.     "  It  is  one  of  the 
most  successful  specimens  of  the  Italian  style  in  the 
United  States." 

This  villa  is  built  of  brick,  painted  externally 
of  a  light  freestone  color,  and  the  window  dressings, 
string-courses,  cornices,  brackets,  &c.,  are  all  free- 
stone. Figs.  113  and  114  are  the  front  and  side 
elevation  of  the  balcony  window  in  the  front  of  the 
house,  drawn  to  a  scale  of  one-quarter  of  an  inch 
to  a  foot. 

Stables.— Under  this  general  name  are  included 
the  barn,  or  the  receptacle  of  hay  and  fodder,  the 
carriage-house  and  the  stable  proper,  or  lodging- 
house  for  horses  and  cows.  The  first  two  may  be 
included  under  one  roof,  the  carriages  on  the  1st 
floor,  hay  in  the  loft,  and  oats  in  the  cellar ;  but 
the  lodging-place  should  be  distant  in  a  wing  attached 
to  the  barn,  that  the  odors  from  the  animal  may  not 
impregnate  the  food,  or  the  cloth-work  of  the  car- 
riages, or  the  ammonia  tarnish  their  mountings. 

Hay  in  bulk,  in  the  mow,  occupies  about  7  c. 
ft.  per  ton ;  bales  average  2'.-l"  x  2/.6//  x  4/,  and 
weigh,  from  220  to  320  Ibs.  The  door  space  for  a 
load  of  hay  in  the  bulk  should  be  from  12  to  13  ft. 
high  and  12  ft.  wide.  The  floor  beneath  the  hay 
should  be  tight,  so  that  dust  and  seed  may  not  drop 
on  the  carriage.  A  door  for  carriage  should  be  10  ft.  6  in.  high  x  9  ft.  wide. 
The  horse  is  to  be  treated  with  greater  care  than  any  other  domestic 
animal.  His  stable  is  to  be  carefully  ventilated,  that  he  may  have  fresh 
air  without  being  subject  to  cross-drafts.  Preferably  the  floor  should  be  on 
the  ground,  that  there  may  be  no  cold  from  beneath.  He  should  stand  as 
near  as  possible  level ;  and  for  this  purpose  a  grated  removable  floor,  with 
small  interstices,  should  be  laid  over  a  concrete  bottom,  with  a  drip  towards 
the  rear  of  the  stall,  and  the  urine  should  be  collected  in  a  drain,  and  dis- 
charged into  a  trapped  manure-tank  outside  the  stable.  The  manure 
should  never  be  deposited  beneath  the  stable,  but  should  be  wheeled  out, 
and  deposited  in  a  manure-yard  or  tank  daily.  It  is  as  essential  that  all 
excrements  should  be  removed  entirely  from  the  stable  as  that  the  privy 
should  be  placed  outside  the  house. , 


Fig.  114. 


ARCHITECTURAL   DRAWING. 


285 


The  breadth  of  stalls  should  be  from  4  ftfci  6  in.  to  5  ft.  in  the  clear ; 
the  length,  7"  ft.  6  in.  to  8  ft :  the  rack  and  feed-box  require  two  feet  in 
addition,  to  which  access  is  given  in  the  best  stables  by  a  passage  in  front. 
Hack  and  feed-box  are  often  made  of  iron,  and  the  upper  part  of  stalls 
fitted  with  wrought-iron  guards.  Box-stalls,  in  which  horses  are  shut  up, 
but  not  tied,  in  cases  of  sickness  or  foaling,  are  about  10  ft.  square. 

Grain  and  hay  are  delivered  from  the  loft  to  the  stable-floor  by  shoots, 
or  boxes  with  slides. 


in  PH. 


tin 


Fig.  115. 

Fig.  115  is  one-half  the  second-story  plan,  and  Plate  LXX.  an  eleva- 
tion in  perspective  of  a  tenant-house,  built  after  designs  by  John  "W. 
Hitch,  architect. 

In  its  construction  it  is  almost  entirely  fire-proof;  the  staircases  are  of 
iron,  the  hall  floors  are  constructed  with  iron  beams,  brick  arches,  and  blue 
flagging ;  the  dividing  floors  and  walls  are  deafened ;  every  alternate  wall 
is  of  brick ;  every  window  has  inside  shutters,  and  every  room  is  ventilated 
by  air-flues  to  the  roof. 

On  the  south  is  a  spacious  flagged  court-yard,  12  by  188  feet,  which  is 
used  by  the  inmates  for  washing  and  drying  their  clothes — the  families 
on  each  floor  having  the  exclusive  use  of  it  for  specified  days  of  the  week. 
The  yard  connects  with  the  main  hall,  by  cross-halls,  and  is  shut  oft'  from 
the  streets  by  high  gates  that  are  kept  closed,  except  fuel  is  brought  to  the 
premises.  The  cellar  is  divided  into  94  compartments,  that  is,  one  for 
each  tenement,  with  a  lock  and  key  for  each. 

On  the  upper  floor  are  two  large  adjoining  rooms,  53'  x  50'  each, 
which  can  be  thrown  into  one  or  disconnected  at  pleasure.  They 
are  designed  for  lectures,  concerts,  or  moral  and  educational  uses  for 


286 


ARCHITECTURAL   DRAWING. 


the  inmates  during  the  wee*:,  and  for  Sunday  school  and  religious  observ- 
ances on  the  Sabbath. 

The  exterior  is  of  brick  with  brown  stone  window-sills,  and  in  its  style 
Is  an  excellent  example  of  the  architectural  effect  that  may  be  produced 
in  our  most  common  materials,  and  in  an  unpretending  edifice,  by  break- 
ing up  the  monotony  of  fagade  by  even  slight  projections,  by  the  clustered 
and  circular  heads  of  the  windows,  and  by  an  appropriate  and  varied 
cornice.  This  style  is  becoming  very  popiilar  and  is  particularly  appli- 
cable to  the  construction  of  mills  and  workshops. 

Store  and  Warehouses. — Plate  LXX1.,  is  an  elevation  of  a  store  front, 
and  figs.  117  and  118  plans  of  first  story  and  basement. 


These  plans  may  be  taken  as  a  type  of  the  general  class  of  large  whole- 
sale or  retail  stores  covering  but  one  lot.  In  this  city  there  is  usually 
beneath  the  sidewalks  two  stories,  the  basement  and  sub-cellar.  These 
are  generally  let  with  the  first  story,  and  the  upper  stories  together  by 
themselves.  The  depth  of  the  stores  are  mostly  from  100  to  200  feet,  on 
an  average  about  150  feet.  The  centre  is  lighted  by  a  skylight  in  the  roof, 
and  by  well-holes,  B,  beneath,  in  the  several  floors.  In  front  of  the  en- 
trance is  a  platform,  A,  which  is  either  an  iron  grating,  or,  when  the  base- 
ment extends  through  into  the  front  vaults,  covered  with  patent  vault 
lights.  To  protect  the  vaults  from  moisture  the  walls  are  laid  hollow,  and 
the  outside  covered  with  asphalte.  The  hoistway  t6\  basement  and  sub- 
cellars,  is  by  a  trap  in  the  grating  front  of  the  window,  usually  a  plat- 
form supported  by  chains  at  the  four  corners,  and  raised  vertically,  often 


ARCHITECTURAL    DRAWING.  287 

by  a  car  sliding  on  an  incline,  if  there  are  outisde  stairs  leading  to  the 
basement.  In  the  rear  an  area  of  some  ten  to  fifteen  feet  in  width  is  dug 
out,  and  the  two  lower  stories  show  full.  All  the  rear  windows  are  pro- 
tected by  iron  shutters. 

The  floor  of  the  first  story  is  often  laid  with  a  rising  grade,  of  about 
1  foot  in  100  towards  the  rear,  to  prevent  the  appearance  which  a  long 
level  sometimes  has  of  descending,  and  to  afford  more  light  in  the  rear  to 
the  basement.  The  offices  are  in  the  rear  on  this  floor.  The  safe  is  some- 
times built  into  the  wall,  or  into  a  projection  from  it,  or  the  safe  is  mova- 
ble ;  or,  what  is  rare  at  present,  a  book  vault  is  made  in  the  front  vault. 
The  front  windows  and  doors  are  mostly  protected  by  revolving  shutters 
rolling  up  like  a  curtain  in  the  box  lintels  above.  Separations  are  made 
between  tenants  occupying  different  floors  by  iron  framed  skylights  over 
the  well-holes. 

C  is  the  entry  way  to  the  second  story,  separated  from  the  store  by  a 
glass  partition  protected  by  a  wrought  iron  screen  or  guard.  Above  this 
entrance  in  the  second  floor,  is  the  hoistway  for  goods,  generally  about  five 
feet  square.  The  second  floor  does  not  differ  in  plan  from  the  first,  and 
so  with  the  stories  above,  except  in  some  cases  the  well-holes  are  wider 
in  the  upper  stories.  The  floors  are  all  level. 

The  water  closets  are  mostly  on  the  third  floor,  and  in  the  front  base- 
ment vault.  The  heating  is  either  by  stoves,  hot  air  furnaces,  or  steam. 
The  shelvings,  counters  and  other  furniture  depend,  of  course,  on  the  class 
and  kind  of  business. 

Front  Elevation. — Various  styles  are  adopted,  but  in  one  particular 
there  is  almost  an  uniformity ;  that  is,  the  whole  front  is  supported  on  posts 
of  cast  iron  in  the  first  story,  with  iron  lintels  and  cornice ;  the  great  ob- 
ject being  to  get  as  much  light  as  possible  in  this  story.  These  posts  are 
sometimes  square  or  rectangular  in  plan,  with  a  small  sunk  panel  on  the 
face,  and  shield-like  ornaments  containing  the  number  of  the  store,  and 
capitals  at  the  top  ;  sometimes  a  sort  of  Corinthian  column,  and  some- 
times two  posts,  the  inside  one  circular,  and  the  outside  square.  As  there 
is  but  little  chance  for  ornament,  the  building  seldom  assumes  any  distinc- 
tive expression  till  it  reaches  the  second  story.  The  great  ornament  of 
the  first  story  is  the  plate  glass.  The  elevation  and  plans  represent  the 
usual  form  of  the  wholesale  stores  with  but  three  openings  in  the  first  story 
— one  window  and  two  doors.  In  the  retail  stores  occupying  a  full  lot 
there  are  generally  four  openings,  the  door  to  the  first  floor,  central  between 
two  windows,  and  the  side  door  leading  to  the  second  story ;  but  where 
all  the  stories  are  occupied  by  the  same  trade,  the  side  door  is  usually 


288  ARCHITECTURAL   DRAWING. 

omitted.  The  door  of  the  retail  store  is  generally  recessed,  with  show 
windows  at  the  sides  to  admit  of  the  greater  display  of  goods.  The  glass 
of  the  windows  are  sometimes  of  one  plate,  as  large  as  8  x  14  feet  even, 
but  more  usually  in  four  squares ;  seldom  more  in  number. 

Above  the  first  story,  the  front  begins  to  assume  an  architectural  ex- 
pression, though  seldom  perhaps  very  significant  of  any-intention  or  de- 
sign for  a  specific  purpose  inside.  The  example  selected  may  be  con- 
sidered a  fair  average  of  the  class.  It  is  to  be  remarked  that  where 
various  businesses  are  to  be  carried  on  in  the  same  building,  and  where 
large  signs  may  be  necessary  to  designate  them,  there  will  be  but  little 
room,  as  there  will  but  little  necessity,  for  much  ornamental  detail. 

Plate  LXXII.  is  a  chaste  and  beautiful  fagade  of  two  stores,  erected 
on  Broadway,  from  designs  by  J.  B.  Snook,  architect. 

Plate  LXXIII.  is  an  elevation  of  a  storefront  executed  in  cast  iron  byD. 
D.  Badger  &  Co.  of  this  city.  The  style  is  Venetian,  and  when  the  front  is 
more  than  fifty  feet  in  width,  the  effect  is  imposing.  It  is  rather  more  ap- 
propriate for  stores  with  offices  above,  or  for  stores  designed  for  but  one 
purpose,  as  signs  larger  than  could  be  placed  in  the  panels  would  mar  the 
effect.  Iron  was  first  introduced  for  house  fronts  by  Mr.  Bogardus,  and 
it  has  much  to  recommend  it.  Ornaments  can  be  applied  profusely,  and 
at  the  same  time  cheaply,  and  in  durability  it  exceeds  our  common  free- 
stones. The  chief  objection  at  present  lies  in  this,  that  few  wish  to  go  to 
the  expense  of  new  patterns :  the  result  is  that  the  forms  become  too  ste- 
reotyped, especially  objectionable  when  much  ornament  is  used.  The  color 
which  it  should  be  painted  has  been  a  subject  of  much  discussion  ;  the 
prevailing  tint  at  present  is  a  sort  of  cream  color,  with  brown  trimmings 
of  the  windows. 

School  Houses. — PL  LXXIY.  contains  a  plan  and  elevation  of  a  dis- 
trict school  house,  with  seats  for  forty-eight  scholars.  There  are  two  en- 
trances, one  for  each  sex,  with  ample  accommodations  of  entry  or  lobby 
room  for  the  hanging  up  of  hats,  bonnets  and  cloaks.  A  side  door  leads 
from  each  entry  into  distinct  yards,  and  an  inside  door  opens  into  the 
school-room.  The  desk,  T,  of  the  teacher,  is  central  between  the  doors, 
on  a  platform,  P,  raised  some  six  or  eight  inches  above  the  floor.  In 
the  rear  of  the  teacher's  desk  is  a  closet  or  small  room,  for  the  use  of 
the  teacher.  The  seats  are  arranged  two  to  each  desk,  with  two  alleys 
of  eighteen  inches,  and  a  central  one  of  two  feet ;  the  passages  around 
the  room  are  three  feet.  The  scale  is  eight  feet  to  the  inch.  The  eleva- 
tion is  in  a  very  plain  Romanesque  style,  to  be  constructed  of  brick  with 
hollow  walls. 


AKCIIITECTUEAL   DRAWING.  289 

On  the  Requirements  of  a  School-House. — Every  scholar  should  have 
room  enough  to  sit  at  ease,  his  seat  should  be  of  easy  access,  so  that  he 
may  go  to  and  fro,  or  be  approached  by  the  teacher  without  disturbing 
any  one  else.     The  seat  and  desk  should  b.e  properly  proportioned  to  each 
other  and  to  the  size  of  the  scholar  for  whom  it  is  intended.     The  seats  ast 
furnished  by  the  different  makers  of  school  furniture,  vary  from  nine  to 
sixteen  inches  in  height ;  and  the  benches  from  seventeen  to  twenty-eight 
inches ;  measuring  on  the  side  next  the  scholar.     The  average  width  of  the 
desk  is  about  eighteen  inches,  and  is  formed  with  a  slope  of  from  one  and 
a  half  to  two  and  a  half  inches,  with  a  small  horizontal  piece  of  from  two 
to  three  inches  at  top.     There  is  a  shelf  beneath  for  books,  but  it  should 
not  come  within  about  three  inches  of  the  front.     The  width  of  the  seat 
varies  from  ten  to  fourteen  inches,  with  a  sloping  back,  like  that  of  a  chair ; 
it  should,  in  fact,  be  a  comfortable  chair.     It  will  be 
observed  that,  in  the  plate,  two  scholars  occupy  one    [     p  I     p  j~~p  f] 
bench ;  fig.  119  represents  another  arrangement,  in 
which  each  scholar  has  a  distinct  bench  ;    and,  in         h       rj  !      h 
many  respects  it  is  preferable,  but  is  not  quite  so  —     ' — '     " 

economical  in  room.     In  primary  schools,  desks  are    [     L— ,  I     L— ,  I     L— ,  f| 

not  necessary ;    and  in  many  of  the   intermediate    | H  I H  I \     II 

schools,  the  seat  of  one  bench  is  formed  against  the    j — i     i — i     i — i     rt 
back  of  the  next  bench  ;  but  distinct  seats  are  pre-    I     U  IP       P 
ferable.     The  teacher's  seat  is  invariably  on  a  raised  rig.  im 

platform,  and  had  better  be  against  a  dead  wall  than  where  there  are  win- 
dows. The  best  light  is  undoubtedly  a  skylight,  but  as  this  is  seldom  con- 
venient, the  lights  at  the  side  should  be  high  above  the  floor.  Blackboards 
and  maps  should  be  placed  along  the  walls:  Care  should  be  taken  in  the 
warming  and  ventilation  ;  the  room  should  not  be  less  than  ten  feet  high ; 
the  best  method  of  heating  is  by  furnaces  in  the-  cellar,  warm  air  should 
be  introduced  in  proportion  to  the  number  of  scholars,  and  ventiducts 
should  be  formed  to  carry  off  the  impure  air. 

In  cities  the  school-houses  are  made  of  a  number  of  stories — the  pri- 
maries being  in  the  lower  stories,  and,  in  some  cases,  play  rooms  also,  and 
the  grammar-schools  occupying  floors  above.  In  these  cases  the  teachers 
are  numerous,  and  separate  rooms  are  prepared  for  the  hearing  of  recita- 
tions. 

Plate,  page  290,  is  a  view  of  one  of  the  largest  of  the  New  York 
City  schools,  of  which  fig.  1,  p.  291,  is  the  plan  of  grammar-department 
floors,  and  fig.  2  plan  of  the  same  floors  of  another  house  of  a  different 
outline. 


290 


ARCHITECTURAL   DRAWING. 


AKCHITECTUKAL   DKAWISG. 


291 


Fig.  1. 


Fig.  2. 


D:0:C:G:Q:D:Q:0:0:0:C:Q: 


g 


CUSS   I         I   ROOM 


a  c± 


CUSS   |  |    RQUN 


SCHOOL  ROO 


D:D:D:Q:C:D:D:D:D:0:D:D:D:D:D: 
0:0:0:0:rj:[:D:fl:G:Q:fl:C:0:C:0: 


RDROB 


i. 


r  J-gCE^J 

.CUSS   . 1      ROC 


CT3    CZ3 
<=? 


292  ARCHITECTURAL   DE AWING. 

Lecture  Rooms,  Churches,  Theatres,  Legislative  Halls.-— To  the  proper 
construction  of  rooms  or  edifices  adapted  for  these  purposes  some  know- 
ledge of  the  general  principles  of  acoustics,  and  their  practical  application, 
is  necessary.  In  the  case  of  lecture  rooms  and  churches,  the  positions  of 
the  speaker  and  the  audience  are  fixed ;  in  theatres,  one  portion  of  the 
enclosed  space  is  devoted  to  numerous  speakers,  and  the  other  to  the 
audience ;  in  legislative  halls,  the  speakers  are  scattered  over  the  greater 
part  of  the  space,  and  also  form  the  audience. 

The  transmission  of  sound  is  by  vibrations,  illustrated  by  the  waves 
formed  by  a  stone  thrown  into  still  water ;  but  direction  may  be  given  to 
sound,  so  that  the  transmission  is  not  equally  strong  in  every  direction ; 
thus,  Saunders  found  that  a  person  reading  at  the  centre  of  a  circle  of  one 
hundred  feet  in  diameter,  in  an  open  meadow,  was  heard  most  distinctly 
in  front,  not  as  well  at  the  sides,  but  scarcely  at 
all  behind.  Fig.  120  shows  the  extreme  dis- 
tance every  way  at  which,  the  voice  could  be 
distinctly  heard :  ninety-two  feet  in  front,  seventy- 
five  feet  on  each  side,  and  thirty-one  feet  in  the 
rear.  The  waves  of  sound  are  subject  to  the 
same  laws  as  those  of  light,  the  angles  of  reflec- 
tion are  equal  to  those  of  incidence ;  therefore, 
in  every  enclosed  space,  there  are  reflected  sounds 
more  or  less  distinct,  according  to  the  position  of -the  hearer,  and  to  the  form 
and  condition  of  the  surfaces  against  which  the  waves  of  sound  impinge. 
Thus,  of  all  the  sounds  entering  a  parabolic  sphere,  the  reflected  sounds 
are  collected  at  the  focus.  Solid  bodies  reflect  sound,  but  draperies  absorb 
it.  As,  in  all  rooms,  the  audience  can  never  be  concentrated  at  focal 
points,  nor  is  it  possible  in  any  construction  to  make  calculation  for 
all  positions,  it  is  in  general  best  to  depend  on  nothing  but  the  direct  force 
of  the  voice,  and  not  to  construct  larger  than  can  be  heard  directly  without 
aids  from  reflected  sounds. 

There  is  great  difference  in  the  strength  of  voice  of  different  speakers  ; 
the  limits  as  given  in  the  figure  are  for  ordinary  reading  in  an  open  space. 
In  enclosed  spaces,  owing  to  the  reflected  sounds  or  some  other  cause,  there 
are  certain  pitches  or  keys  peculiar  to  every  room,  and  to  speak  with  ease, 
the  speaker  must  adapt  his  tone  to  those  keys.  The  larger  the  room,  the 
slower  and  more  distinct  should  be  the  articulation. 

It  has  been  observed,  that  the  direction  of  the  sound  influences  the  ex- 
tent to  which  it  may  be  heard.  The  direction  of  the  currents  of  air  through 
which  the  sound  passes  effects  the  transmission  of  the  sound,  and  this  may 


ARCHITECTURAL   DRAWING.  293 

be  made  useful  when  the  rooms  are  heated  by  hot  air,  by  introducing  the 
air  near  the  speaker,  and  placing  the  ventilators  or  educts  at  the  outside 
of  the  rooms,  and  by  placing  their  apertures  rather  nearer  the  bottom 
of  the  room  than  at  the  top.  It  would  seem  much  better  and  easier 
to  make  a  current  of  air  a  vehicle  of  sound  rather  than  depend  on  re- 
flection. 

The  best  form  for  a  lecture  room  is  the  semicircle,  or  three-fifths  of  a 
circle,  fig.  120,  the  speaker  in  the  one  case  at  the  centre,  in  the  other,  at 
the  point  A,  on  a  platform  raised  some  two  or  three  steps  above  the  floor, 
the  audience  being  ranged  in  concentric  seats,  rising  from  the  centre 
outwards.  The  room  should  be  no  higher  than  requisite  for  beauty 
or  for  ventilation.  The  ceiling  should  be  slightly  curved,  not  flat  nor  half 
globe. 

On  the  space  occupied  by  seats  in  general. — A  convenient  arm-chair  oc- 
cupies about  twenty  inches  square,  the  seat  itself  being  about  eighteen 
inches  in  depth,  and  the  slope  of  the  back  two  inches.  Eighteen  inches 
more  affords  ample  space  for  passage  in  front  of  the  sitter :  this  accommoda- 
tion would  be  ample.  In  the  arrangement  of  seats  at  the  Academy  of  Mu- 
sic the  bottom  turns  up,  and  twenty-nine  inches  only  is  allowed  for  both  seat 
and  passage,  and  eighteen  inches  for  the  width  of  seat,  which  may  be  taken 
as  the  average  allowance  in  width  to  each  sitter  in  comfortable  public 
rooms.  In  lecture  rooms  stalls  are  often  used,  the  space  there  occupied  by 
seat  and  passage  being  about  two  feet  six  inches.  The  alleys  should 
be  at  the  sides  of  the  room,  with  two  intermediate,  dividing  the  seats 
into  three  equal  benches,  and  not  one  in  the  centre,  except  in  very, 
large  rooms,  as  the  space  thus  left  is  the  best  for  hearing  and  seeing  the 
speaker. 

In  the  earlier  churches,  ceremonies  and  rites  formed  a  very  large  part 
of  the  worship,  the  sight  was  rather  appealed  to  than  the  hearing,  and  for 
this  purpose,  churches  were  constructed  of  immense  size,  and  with  all  the 
appliances  of  ornament  and  construction,  with  pillars,  vaults,  groins  and 
traceried  windows.  In  the  churches  of  this  country,  the  great  controlling 
principle  in  the  construction  of  a  church,  is  its  adaptation  to  the  comfort- 
able hearing  and  seeing  the  preacher.  In  this  view  alone,  the  church  is 
but  a  lecture  room :  but  since  even  the  character  of  the  building  may  tend 
to  devotional  feelings  in  the  audience,  and  since  certain  styles  and  forms 
of  architecture  have  long  been  used  for  church  edifices,  and  seem  particu- 
larly adapted  for  this  purpose,  it  has  been  the  custom  to  follow  these  time- 
honored  examples,  adapting  them  to  the  modern  requirements  of  church 
worship. 


294 


ARCHITECTURAL   DRAWING. 


Fig.  121,  is  a  plan  of  an  ancient  basilican   or  Romanesque  church ; 
fig.  122,  a  sectional  elevation  of  the  same.     Fig.  123  is  a  plan  of  a  Gothic 


church  in  which  C  is  the  chancel,  usually  at  the  eastern  extremity,  TT  the 
transept,  and  N"  the  nave.  In  general  elevation  the  Gothic  and  Roman- 
esque agree  ;  a  high  central  nave  and  low  side  aisles.  In  the  later  Roman- 
esque'the  transept  is  also  added. 

The  basilicas  aggregated  within  themselves  all  the  offices  of  the  Rom- 
ish church.  The  circular  end  or  apex,  and  the  raised  platform,  or  dais  in 
front  of  it,  was  appropriated  entirely  to  the  clergy ;  beneath  was  the  crypt 
or  confessional  where  were  placed  the  bodies  of  the  saints  and  martyrs, 
and  pulpits  were  placed  in  the  nave,  from  which  the  services  were  said  or 
sung  by  the  inferior  order  of  clergy. 

The  plan,  fig.  123,  is  that  of  the  original  Latin  cross,  the  eastern  limb 
or  chancel  being  the  shortest,  and  the  nave  the  longest.  Sometimes  the 
eastern  limb  was  made  equal  to  that  of  the  transepts,  sometimes  even  lon- 
ger, but  never  to  exceed  that  of  the  nave.  In  the  Greek  cross  all  the 
•limbs  are  equal.  In  most  of  the  French  Gothic  churches  the  eastern  end 
is  made  semicircular,  often  enclosed  by  three  or  more  apsidal  chapels,  that 
is,  semi-cylinders,  surmounted  by  s^mi-domes. 

The  Byzantine  church  consisted  internally  of  a  large  square  or  rectan- 
gular chamber,  surmounted  in  the  centre  by  a  dome,  resting  upon  massive 
piers ;  an  apse  was  formed  at  the  eastern  end.  Circular  churches  were 
built  in  the  earlier  ages  for  baptisteries,  and  for  the  lombs  of  saints  and  em- 
perors. 

Having  thus  briefly  treated  of  the  general  form  of  ancient  churches, 
we  proceed  now  to  the  consideration  how  far  they  may  be  applied  to 
the  requirements  of  modem  church  services.  The  prime  necessities  are 
those  of  the  lecture  room ;  comfortable  seats,  convenient  for  hearing  and 
seeing  the  preacher ;  and  proper  provision  for  ventilation.  In  addition,  an 
eligible  position  for  the  choir,  a  small  withdrawing  room  for  the  clergy- 
man, and  a  room  suitable  for  Sunday  Schools  and  for  parish  meetings. 


AECHITECTUKAL   DRAWING.  295 

Seats  are  arranged  by  pews  or  stalls,  the  width  of  each  pew  being  in  gen- 
eral about  two  feet  ten  inches.  The  length  of  pews  is  various,  being  gen- 
erally of  two  sizes,  adapted  to  either  small  or  large  families,  say  from  seven 
feet  six  inches,  to  eleven  feet  six,  eighteen  inches  being  allowed  for  each  sit- 
ter. In  arrangement  it  is  always  considered  desirable  that  there  should  be  a 
central  aisle,  and  if  but  four  rows  of  pews,  two  aisles  against  the  wall ;  if 
six  rows,  one  row  on  each  side  will  be  wall  pews.  Few  churches  are  now 
without  an  organ ;  its  dimensions  should  of  course  depend  on  the  size  of 
the  church.  In  form  it  may  be  adapted  somewhat  to  the  place  which  may 
be  appropriated  to  it.  In  general  it  is  oblong  in  form,  the  longer  side  be- 
ing with  the  keys.  The  dimensions  suited  to  a  medium  sized  church  are 
about  nine  feet  by  fifteen,  and  twelve  feet  in  height.  The  withdrawing 
room  for  the  clergyman  may  be  but  of  very  small  dimensions,  and  should 
be  accessible  from  without.  The  Sunday  School,  in  general,  requires  in 
plan  about  half  the  area  of  the  church. 

As  city  residences  differ  from  those  in  the  country,  from  the  same  ne- 
cessities do  the  city  churches  differ  from  the  rural  ones.  A  very  common 
form  of  city  church  is,  in  plan,  that  of  the  Latin  cross,  with  extremely 
short  transepts  and  chancels ;  sometimes  the  roof  is  supported  by  pillars, 
with  imitated  vaults  in  plaster,  but  often  with  a  double  pitch  roof,  and 
open  timber  finish  in  the  inside.  The  organ  loft  is  sometimes  in  one  of 
the  transepts,  sometimes  at  the  back  of  the  congregation  over  the  door  of 
entrance. 

A  sort  of  basilican  church  is  also  very  common :  rectangular  in  form 
with  a  small  semicircular  niche  behind  the  preacher,  and  small  withdraw- 
ing rooms  or  vestries  at  each  side  of  it.  The  ceilings  are  finished  after  the 
Greek  style,  with  sunk  panels,  sometimes  coved,  with  pilasters  but  seldom 
pillars,  except  short  ones,  to  support  the  galleries  which  are  adopted  in 
this  style  of  buildings,  but  not  so  commonly  in  the  Gothic.  The  rooms  for 
Sunday  Schools  are  almost  invariably  in  the  basement  of  the  city  churches. 

The  basilican  form  is  evidently  the  most  economical  in  its  occupation 
of  land ;  if  the  church  be  situated  at  the  corner  of  two  streets,  it  can  cover 
the  whole  lot,  one  side,  or  a  portion  of  one  side  being  left  blank  of  win- 
dows. If  an  elevation  similar  to  fig.  122  be  adopted,  the  light  can  be  ta- 
ken in  at  clere-story  windows.  But  this  form  is  objectionable  as  requiring 
pillars  in  construction,  which,  unless  made  of  iron,  and  of  small  size,  very 
much  interfere  with  sight  and  hearing. 

The  position  of  our  city  churches  is  usually  as  we  have  said  at  the  cor- 
ner of  streets,  but  if  they  can  be  placed  so  far  in  the  centre  of  a  lot  as  to 
receive  the  light  from  the  back  areas,  the  position  is  preferable  as  removed 


296  ARCHITECTURAL   DRAWING. 

from  the  noise  of  passing  vehicles.  In  that  case  the  church  proper  is  ap- 
proached by  a  long  aisle,  above  which  may  be  the  room  for  the  Sunday 
School.  This  room  should  be  fitted  with  water-closets,  in  fact  they  would 
be  often  of  great  convenience  connected  with  all  churches. 

In  elevation,  city  churches  are  Greek  with  porticoes  in  front,  Roman- 
esque and  Gothic,  occasionally  Byzantine.  The  Greek  have  no  tower  but 
often  a  spire  above  the  portico ;  the  Romanesque  and  Gothic  generally  one 
tower,  over  the  central  door  of  entrance,  or  at  one  corner ;  sometimes  two, 
one  at  each  side  of  the  principal  door,  almost  invariably  surmounted  by 
spires,  high  and  tapering,  usually  of  wood,  but  in  some  instances  of  stone. 

Plate  LXXV.,  is  a  design  for  a  church  in  the  English  Decorated 
Gothic  style.  It  will  be  observed  in  the  design  that  there  is  a  side  en- 
trance with  its  appropriate  gable ;  in  a  similar  way,  small  edifices  may  be 
attached  to  the  main  one,  for  necessary  offices,  parsonages,  or  Sunday 
schools,  adding  much  to  the  picturesque  effect,  and  particularly  appropri- 
ate to  country  churches. 

Plate  LXXVI.  is  the  original  design  (by  James  Renwick,  architect)  of 
the  front  elevation  of  the  Roman  Catholic  cathedral  now  being  built  in 
Fifth  avenue.  The  style  is  the  French  Decorated  Gothic,  and  the  faQade 
is  more  extensive  than  that  in  process  of  construction. 

Plate  LXXYII.  is  a  design  in  the  Romanesque  or  Byzantine  style,  by 
Messrs.  Renwick  &  Sands,  architects.  It  is  now  being  built  in  Fourth 
avenue ;  but  the  drawing  is  incomplete,  being  partially  a  working  one, 
and  a  campanile  forms  a  part  of  the  design. 

It  is  the  custom  in  Episcopal  churches  to  place,  if  possible,  the  chancel 
at  the  eastern  end,  and  often  a  large  window  at  the  extremity  of  the  chan- 
cel. The  light  from  this  window  should  be  very  much  subdued,  as  it  comes 
full  in  the  eyes  of  the  congregation ;  for  the  comfort  of  preacher  and  peo- 
ple a  side  or  top-light  in  the  chancel  would  be  much  better.  A  very  beau- 
tiful effect  is  produced  by  skylight  in  the  apse  of  the  Romanesque  church, 
which,  being  high  above  the  congregation,  does  not  interfere  with  them, 
and  affords  the  best  light  to  lead  the  services.  The  light  in  churches 
should  not  be  garish,  but  subdued  and  well  diffused,  which  will  be  best 
effected  by  light  from  windows,  placed  as  high  up  as  possible.  A  single 
north  window,  in  many  small  churches,  would  be  sufficient  for  all  purposes, 
would  not  injure  the  eyes  of  the  congregation  by  cross  lights,  would  add 
very  much  to  the  effect  when  the  walls  are  painted  in  fresco  or  distemper, 
and,  if  suitable  means  are  provided  for  summer  ventilation,  no  other  windows 
would  be  necessary.  In  some  recent  city  churches  the  light  in  the  daytime 
is  taken  entirely  from  skylights,  and  at  night  from  gas  lights  placed  in 


ARCHITECTURAL -DKAWING.  297 

the  roof  of  the  church,  and  reflected  below  through  the  same  apertures  in 
the  ceiling. 

Theatre. — The  requirements  of  theatres  and  opera  houses,  differ  essen- 
tially from  those  of  lecture  rooms  and  churches,  in  that  the  audience 
themselves  form  an  important  part  of  the  exhibition.  It  is  not  only  ne- 
cessary that  the  audience  should  have  a  good  position  for  hearing  and  see- 
ing the  performance  upon  the  stage,  but  also  to  see  each  other.  The  most 
approved  form,  now,  for  the  body  of  the  house,  is  a  circular  plan,  the 
opening  for  the  stage  occupying  from  one-fourth  to  one-fifth  of  the  circum- 
ference, the  sides  of  the  proscenium  being  short  tangents.  The  circular 
form  is  well  adapted  for  both  hearing  and  seeing,  and  also  for  lighting. 
»  In  the  general  position  of  the  stage,  proscenium,  orchestra,  orchestra 
seats,  parquette,  and  boxes,  but  one  plan  is  followed.  "We  proceed  to  give 
briefly  the  usual  arrangements  of  seats,  and  some  other  requirements,  and  a 
small  table  of  the  proportions  of  different  houses.  The  line  of  the  front  of 
the  stage,  at  the  foot  lights,  is  generally  slightly  curved,  with  a  sweep,  say, 
equal  to  the  depth  of  the  stage,  and  the  orchestra  and  parquette  seats  are 
arranged  in  circles  concentric  with  it :  of  the  space  occupied  by  seats  we  have 
already  spoken.  The  entrance  to  the  parquette  may  be  through  the  boxes, 
near  the  proscenium,  and  often  centrally,  but  better  at  the  sides,  dividing  the 
boxes  into  three  equal  benches ;  the  seats  in  the  boxes  are  usually  concentric 
with  the  walls,  and  more  roomy  than  those  of  the  parquette.  The  orches- 
tra seats  are  of  a  height  to  bring  the  shoulders  of  the  sitter  level  with  the 
floor  of  the  stage,  and  the  floor  of  the  parquette  rises  to  the  outside,  1  in 
15  to  18.  The  floor  of  the  first  row  of  boxes  is  some  2  to  3  feet  above  the 
floor  of  the  parquette  at  the  front  centre,  and  rises  by  steps  at  each  row, 
some  4  inches  ;  in  the  next  tier  of  boxes  the  steps  are  considerably  more 
in  height,  and  so  on  in  the  boxes  above.  In  general,  three  rows  of  boxes 
are  all  that  is  necessary  ;  in  front,  above  the  second,  the  view  of  the  stage 
is  almost  a  bird's  eye  view.  The  floor  of  the  stage  descends  to  the  foot- 
lights at  the  rate  of  about  1  in  50.  In  large  theatres  it  is  of  the  utmost 
importance  that  all  the  lobbies  or  entries  should  be  spacious,  and  the 
means  of  exit  numerous  and  ample.  The  staircases  broad,  in  short  flights 
and  square  landings,  and  not  circular,  as,  in  case  of  fright,  the  pressure  of 
persons  behind  may  precipitate  those  in  front  the  whole  length  of  the 
flight.  Ladies'  drawing  rooms  should  be  placed  convenient  to  the  lobbies, 
of  a  size  adapted  to  that  of  the  theatre,  arranged  with  water  closets ; 
there  should  also  be  provided  rooms  for  the  reception  of  gentlemen's  canes 
and  umbrellas,  with  water  closets  attached.  The  box-office  should  be,  of 
course,  near  the  entrance,  but  so  arranged  as  to  interfere  as  little  as  possi- 


298 


ARCHITECTURAL   DRAWING. 


ble  with  the  approach  to  the  doors  of  the  house.  At  the  entrance  there 
should  be  a  very  spacious  lobby,  or  hall,  so  that  the  audience  may  wait 
sheltered  against  the  weather ;  if  possible,  there  should  be  a  long  portico 
over  the  sidewalk,  to  cover  the  approach  to  the  carriages.  But  single  en- 
trances are  necessary  to  distinct  parts  of  the  house,  but  the  greater  the 
number  of,  and  the  more  ample  places  for  exit,  at  the  conclusion  of  the 
piece,  the  better. 

COMPAEATIVE  TABLE  OF  THE  DIMENSIONS  OF  A  FEW  THEATRES. 


DISTANCE,   IX  FEET. 

HEIGHT,  IN  FEET. 

NAiffi  AXD  LOCATION. 

Bet.  boxes  and 
foot-lights. 

l! 
fl 

Bet.  curtain  and 
back  of  stage. 

Create  at  breadth 
of  pit. 

•s 

f  .3 

11 

£  S 

P3 

Breadth  of  stage 
but.  eiUo  walls. 

From  floor  of  pit 

ti 
=1 

li 

Is 

Alexandra,  St.  Petersburg, 

65 

11 

84 

53 

56 

75 

53 

58 

,  Berlin,       

62 

16 

76 

51 

41 

92 

43 

47 

La  Scala,  Milan,    

77 

IS 

73 

71 

49 

86 

CO 

64 

San  Carlo,  Naples,       .... 

77 

18 

74 

74 

52 

66 

81 

83 

Grand  Theatre,  Bordeaux, 

46 

10 

69 

47 

37 

80 

50 

57 

Salle  Lepellctier,  Paris, 

67 

9 

82 

66 

43 

73 

52 

66 

Co  vent  Garden,  London, 

66* 

55 

51 

32 

86 

54 

Drury  Lane,              " 

64* 

80 

56 

32 

48 

60 

Boston,  Boston,    

53 

18 

63 

46 

87 

55} 

53 

Academy  of  Music,  New  York,  . 

74 

13 

71 

62 

48 

83 

74 

Pike's  Opera  House,        " 

54 

8} 

63} 

4S 

44 

76 

52 

67 

Opera  House,  Philadelphia, 

61 

17 

72 

66 

43 

90 

64} 

74 

*  These  dimensions  include  the  distance  between  the  foot-lights  and  curtain. 

Although  much  has  been  written  about  the  construction  of  legislative 
halls,  in  relation  to  acoustic  principles,  there  yet  seems  to  be  great  disa- 
greement in  practical  examples,*and  in  the  deductions  of  scientific  men. 
The  Chamber  of  French  Deputies  was  constructed  after  a  report  of  most 
celebrated  architects,  in  a  semicircular  form,  surmounted  by  a  flat  dome, 
but  as  the  member  invariably  addresses  the  house  from  the  tribune,  at  the 
centre,  in  its  requirements  it  is  but  a  lecture  room.  Mr.  Mills,  Architect, 
of  Philadelphia,  recommends  for  legislative  or  forensic  debate,  a  room  cir- 
cular in  its  plan,  with  a  very  slightly  concave  ceiling.  Dr.  Reid,  on  the 
contrary,  in  reference  to  the  Houses  of  Parliament,  gave  preference  to  the 
square  form,  with  a  low,  arched  ceiling.  The  Hall  of  Representatives  nearly 
completed  at  Washington,  is  139  feet  long,  by  93  feet  wide,  and  about  36 
feet  high,  with  a  spacious  retiring  gallery  on  three  sides,  and  a  reporter's 


ARCHITECTURAL  DRAWING.  299 

gallery  behind  the  Speaker's  chair.  The  members'  desks  are  arranged  in  a 
semicircular  form.  The  ceiling  is  flat,  with  deep  sunk  panels,  openings  for 
ventilation,  and  glazed  apertures  for  the  admission  of  light.  The  ventila- 
tion is  intended,  in  a  measure,  to  assist  the  phonetic  capacity  of  the  Hall, 
the  air  being  forced  in  at  the  ceiling  and  drawn  out  at  the  bottom. 

In  .reviewing  the  general  principles  of  acoustics,  it  will  be  found  that 
those  rooms  are  the  best  for  hearing  in  which  the  sound  arrives  directly  to 
the  ear,  without  reflection ;  that  the  sides  of  the  room  should  not  be  re- 
flectors, not  sounding  boards,  and  that  surfaces  absorbing  sound  are  less 
injurious  than  those  that  reflect.  Slight  projections,  such  as  ornaments  of 
the  cornices  and  shallow  pilasters,  tend  to  destroy  sound,  but  deep  alcoves 
and  recessed  rooms  produce  echoes.  Let  the  ceiling  be  as  low  as  possible, 
and  slightly  arched  or  domed;  all  large  external  openings  should  be 
closed ;  as  M.  Meynedier  expresses  it,  in  his  description  of  an  opera  house, 
"  Let  the  hall  devour  the  sound  ;  as  it  is  born  there,  let  it  die  there." 

Plate  LXXVIII.  is  the  interior  perspective  of  the  New  York  Crystal 
Palace,  an  illustration  of  the  earlier  form  of  a  class  of  buildings  which 
originated  with  Paxton,  in  his  design  for  the  London  Exhibition  of  1851, 
and  of  which  some  of  the  details  of  construction  have  been  already 
given.  These*  buildings  are  composed  wholly  of  iron,  p;lass  and  wood,  but 
no  large  pieces  of  either  material  are  used ;  in  this  consists  their  great  pecu- 
liarity. Stiifness  and  tenacity  of  material  are  applied  rather  than  mass,  to 
counteract  incidental  strains ;  and,  on  this  account,  they  are  not  as  suitable 
as  walls  of  brick  and  stone  for  permanent  structures,  nor  are  they  as  cheap  ; 
and  in  this  respect,  an  improvement  has  been  made  in  the  French  exhi- 
bition building ;  but  for  a  structure  easily  moved  and  put  together,  as  it  was 
intended,  and  for  green  or  hot-houses,  it  seems  especially  adapted ;  and  as 
a  practical  example  of  the  application  of  iron,  and  an  economical  applica- 
tion, it  has  been  of  great  importance. 

Before  concluding  the  article  on  architectural  drawing  it  may  be  ap- 
propriate to  speak  briefly  of  materials  as  applied  in  the  exteriors  of  edifi- 
ces. Sufficient  has  already  been  said  of  their  strength,  we  now  refer 
merely  to  their  fitness  to  architectural  ornament. 

Brick  in  cities  is  by  far  the  most  common  of  all  materials,  nor  do  we 
know,  of  any  more  suited  to  workshops  and  factories,  for  appropriateness, 
economy,  and  durability  (when  hard  burned),  nor  do  we  know  of  any  style 
of  architecture  more  fitted  to  the  material  than  the  Komanesque,  as  in 
Plate  LXX.  Stone  in  the  rough  or  rubble  walls,  laid  in  cement  or  mortar, 
are  often  used  for  these  structures,  but  in  that  case  the  lintels  should  be 
square,  and  if  possible  of  a  different  shade  of  stone. 


300  ARCHITECTURAL   DRAWING. 

For  city  residences,  and  stores,  the  exteriors  are  composed  of  all  sorts 
of  building  materials,  with  the  exception  of  wood,  from  its  insecurity  in 
case  of  fire ;  brick,  with  marble,  freestone,  iron  or  terra  cotta  lintels  and 
sills  for  openings,  red  brick  and  straw-colored  bricks,  brick  on  rusticated 
basements,  and  sometimes  brick  in  alternate  stripes  with  marbles ;  free- 
stone in  a  great  variety  of  shades,  mostly  of  a  reddish  brown,  oftert  fawn 
and  drab ;  marbles  white  and  veined  ;  native  and  foreign  granite ;  and  iron, 
the  use  of  which  in  fronts  is  the  invention  of  our  age,  and  is  destined  to 
modify  our  style  of  architecture. 

All  materials  are  suited  for  country  residences  except  iron;  stone 
houses  may  be  kept  in  their  native  color,  but  brick  or  wood  should  be 
painted.  We  extract  from  Downing  the  following  on  the  color  of  country 
houses.  "  We  think  all  buildings  in  the  country  should  be  of  those  soft 
quiet  shades  called  neutral  tints,  such  as  fawn,  drab,  gray,  brown,  etc.,  and 
that  all  positive  colors,  such  as  white,  yellow,  red,  blue  and  black  should 
always  be  avoided ;  neutral  tints  harmonizing  best  with  nature  and,  posi- 
tive colors  most  discordant. 

In  the  second  place,  we  would  adapt  the  shade  of  color  as  far  as  possible, 
to  the  expression,  style  or  character  of  the  house  itself.  A  Jarge  mansion 
may  receive  a  somewhat  sober,  dignified  hue ;  a  house  of  moderate  size,  a 
lighter  and  more  pleasant  tone ;  small  cottages  should  always  have  a  cheer- 
ful, lively  tint,  not  much  removed  from  white.  Country  houses  thickly 
surrounded  by  trees,  should  always  be  of  a  lighter  shade  than  those  stand- 
ing exposed.  In  proportion  as  a  house  is  exposed  to  view,  let  its  hue  be 
darker ;  and  wbere  it  is  much  concealed  by  foliage,  a  very  light  shade  of 
color  is  to  be  preferred. 

"  A  species  of  monotony  is  produced  by  using  the  same  neutral  tint 
for  every  part  of  the  exterior  of  a  country  house.  A  certain  sprightliness 
is  bestowed  on  a  building  in  neutral  tint  by  painting  the  bolder  projecting 
features  of  a  different  shade.  The  simplest  practical  rule  that  we  can  sug- 
gest for  effecting  this  in  the  most  satisfactory  manner,  is  the  following :  if 
the  tint  selected  for  the  body  of  the  house  be  a  light  one,  let  the  facings 
of  the  windows,  cornices,  etc.,  be  painted  several  shades  darker  of  the 
same  color.  The  blinds  may  either  be  a  still  darker  shade  than  the  fa- 
cings, or  else  the  darkest  green.  If  on  the  other  hand,  the  tint  chosen  is 
a  dark  one,  then  let  the  window  dressings,  etc.,  be  painted  of  a  much 
lighter  shade  of  the  same  color." 

Thus  far  Mr.  Downing.  Most  persons  must  be  struck  with  the  justness 
of  his  remarks  in  general,  but  all  are  not  prepared  entirely  to  ignore  white 
as  a  color  for  country  houses.  We  have  always  fancied  in  contemplating 


ARCHITECTURAL   DRAWING.  301 

an  extensive  landscape  that  jottings  of  white  enlivened  the  scene,  and 
prefer  a  whitewashed  cottage,  carrying  an  air  of  cleanliness,  to  the  least 
admixture  of  neutral  tint :  neither  seems  it  high  art  to  harmonize  always 
with  nature,  it  often  makes  a  very  flat  picture. 

However  we  build,  or  whatever  built  of,  let  the  building  express  the 
purpose,  and  let  the  material  be  suited  to  it.  Xet  those  which  are  intend- 
ed for  time  be  of  lasting  materials,  but  those  that  are  temporary,  be  of 
that  most  convenient ;  let  not  one  imitate  the  other. 

Ventilation  and  Warming. — To  the  proper  construction  of  all  edifices 
some  knowledge  of  the  principles  of  ventilation  and  warming  are  neces- 
sary, as  the  arrangements  for  this  purpose  are  to  be  made  in  planning  the 
building.  Air  is  deteriorated  in  apartments  by  the  respiration  and  perspi- 
ration of  people,  and  by  combustion  in  heating  and  lighting.  At  least  3 
cubic  feet  per  minute  of  fresh  air  should  be  supplied  for  each  person  occu- 
pying the  room,  this  quantity  being  deteriorated  by  respiration  and  per- 
spiration. As  to  combustion,  1  pound  of  carbon  or  charcoal,  in  burning, 
consumes  2.6  pounds  of  oxygen,  which  is  that  contained  in  between  13 
and  14  pounds  of  atmospheric  air ;  and  1  pound  of  hydrogen,  consumes  8 
pounds  of  oxygen,  which  is  that  contained  in  about  40  pounds  of  atmos- 
pheric air.  Now  tallow,  wax  and  oil  contain  upon  the  average  from  YY  to 
80  per  cent  of  carbon,  and  from  11  to  14:  per  cent,  of  hydrogen :  the  per 
centage  of  carbon  in  anthracite  and  bituminous  coal  is  more  various,  but  the 
same  calculations  may  be  used.  100  cubic  feet  of  air  weighs  about  Y 
pounds,  so  from  the  above  data  the  approximate  consumption  of  oxygen 
by  any  given  quantity  of  the  above,  combustibles,  is  easily  calculated. 
The  combustion  of  coal  gas  generally  spoils  thrice  its  bulk  of  oxygen,  or, 
fifteen  times  that  of  air. 

The  ventilation  of  a  building  does  not  necessarily  imply  the  warming, 
but  the  warming  should  include1  means  of  ventilation,  as  ventilation  is  as 
necessary  in  summer  as  winter,  which,  in  warm  weather,  is  mostly  effected 
by  windows  and  -doors ;  but  warming  necessarily  implies  the  closing  of 
these  apertures,  and  the  provision  of  others  through  which  the  passing  air 
may  be  warmed  before  coming  in  contact  with  the  occupants  of  the 
building.  Yentilation  requires  something  besides  large  rooms  and  open 
spaces ;  means  of  circulation  must  be  provided.  A  nuisance  in  a  fenced 
yard,  open  to  the  sky,  may  exist  for  a  long  time  before  the  smell  becomes 
diluted  by  mixture  with  other  air ;  an  attic  without  windows,  in  summer 
would  be  oppressive,  with  all  its  flooring  removed.  Air  may  be  regarded 
as  a  liquid  like  water ;  it  is  moved  more  readily,  but  the  laws  of  its  motion 
are  the  same,  and  for  artificial  movement  or  ventilation  the  application  of 


302  AECHITECTTJKAL   DRAWING. 

heat  is  generally  the  readiest  means.  Air,  as  it  is  heated,  rises,  and  colder 
air  rushes  in  to  supply  its  place ;  if  the  upper  strata  alone  be  heated,  the 
lower  strata  will  still  remain  cool.  Let  the  heat  be  so  applied  that  circu- 
lation is  promoted,  and  that  there  be  as  few  eddies,  or  corners,  as  possible, 
in  which  the  air  may  be  stagnant.  When  heat  is  applied  for  ventilation 
only,  as  in  mines,  a  fire  is  built  in  a  flue  near  the  top,  and  the  air  necessary 
for  combustion  is  drawn  from  the  mines ;  the  flue  extends  from  the  bottom 
of  the  mine,  capped  by  a  chimney  above,  and  ducts  are  led  from  the 
bottom  of  the  flue  to  the  face  of  the  workings,  the  cold  air  for  ventilation 
being  drawn  down  through  the  working-shafts  and  drifts.  Dry  air  is 
almost  a  perfect  non-absorbent  of  heat ;  the  absorbence  is  mostly  due  to 
the  moisture  in  it,  which  acts  also  as  a  governor  or  preserver  of  the  heat, 
preventing  surfaces  from  being  too  readily  heated  or  cooled.  Air  in 
winter  is  very  dry,  but  as  its  volume  is  enlarged  by  heat,  it  draws  a 
supply  of  moisture  from  every  thing  with  which  it  conies  in  contact — from 
the  skin  and  lungs,  creating  that  parched  and  feverish  condition  expe- 
rienced in  many  furnace-heated  houses;  from  furniture  and  wood-work, 
snapping  joints  and  making  unseemly  cracks. 

The  following  table  gives  the  amount  in  grains  Troy  of  moisture  con- 
tained  in  one  cubic  foot  of  air  when  saturated : 


Degrees 
Fahrenheit. 
10                ... 

Grains  of 
in  cubic 
8 

20  

1.3 

30 

2 

40  

2.9 

50              

.     4. 

60 

6 

To  

8. 

80 

10 

90  

15. 

100... 

.  19. 

Thus,  taking  the  air  at  10°,  and  heating  it  to  70°,  the  ordinary  temper- 
ature of  our  rooms  requires  about  9  times  the  moisture  contained  in  the 
original  external  atmosphere,  and,  if  heated  to  100°,  as  most  of  our  hot-air 
furnaces  heat  the  air,  it  would  require  about  23  times. 

Methods  of  Heating. — The  open  fireplace  heats  by  radiation,  commu- 
nicating heat  to  objects  which  by  contact  transfer  it  to  the  air.  Persons 
coming  in  contact  with  rays  are  themselves  heated,  while  the  air  around 
them  is  cool  and  invigorating  for  breathing;  the  bright  glow  has  a 
cheering  and  animating  effect  upon  the  system,  somewhat  like  that  of 


AKCHITECTUKAL   DRAWING.  303 

sunliglit.  As  a  ventilator,  an  open  fire  is  one  of  the  most  important, 
drawing  in  air  not  only  for  the  support  of  combustion,  but  also,  by  the 
heat  of  the  fire  and  flue,  making  a  very  considerable  current  through  the 
throat  of  the  chimney  above  the  fire.  From  this  cause,  although  there  is 
a  constant  change  of  air,  yet  there  arises  one  great  inconvenience  of  disa- 
greeable drafts,  especially  along  the  floor,  if  the  air  supply  be  drawn 
directly  from  the  outer  cold  air ;  but  in  connection  with  properly  regulated 
furnaces  or  stoves,  the  open  fireplace  becomes  the  most  perfect  means  of 
heating  and  ventilation ;  and  here  it  may  be  proper  to  remark,  in  regard 
to  ventilation,  that  a  mere  flue  does  not  imply  that  a  current  will  be  cre- 
ated in  it ;  if  hot,  there  may  be  an  upward  current,  if  cold,  perhaps  the 
reverse,  or,  like  the  neck  to  a  bottle,  with  no  movement  in  it  at  all.  If 
there  be  an  open  fire  in  the  house,  the  air  will  rush  to  supply  it  through 
the  readiest  openings ;  the  parlor  fire  may  be  fed  by  a  current  of  cold  air 
down  staircases  and  well  holes,  and  the  chamber  flues,  or  up  warm  with 
the  smell  from  the  kitchen. 

Stoves. — Open  stoves  heat  by  direct  radiation,  and  by  heating  the  air  in 
contact  with  it ;  close  stoves  by  the  latter  way  only ;  as  economical  means 
of  heating,  they  are  the  best,  and,  when  properly  arranged,  give  both  a 
comfortable  and  wholesome  atmosphere.  There  should  be  some  dish  of 
water  upon  them  to  supply  a  constant  evaporation  of  moisture,  sufficient  to 
compensate  for  increased  capacity  of  the  air  due  to  its  increased  heat.  In 
the  hall  there  will  be  no  objection  to  a  close  stove,  letting  it  draw  its  sup- 
ply of  air  as  it  best  can;  but  in  close  rooms  the  open  stove  is  best,  on  the 
plan  of  the  old  Franklin  stove,  or,  if  a  close  stove,  somewhat  on  the  plan 
of  a  furnace,  with  an  outer  air  supply  for  combustion  and  ventilation,  with 
an  opening  into  the  chimney-flue  at  the  bottom  for  the  escape  of  foul  air, 
and  at  the  top  another  opening  with  a  register  valve  for  the  escape  of  the 
air  when  it  becomes  too  hot. 

Hot-air  furnaces  are  close  cast-iron  stoves,  enclosed  in  air  chambers  of 
brick  or  metal,  intd  which  external  air  is  introduced,  heated,  and  distrib- 
uted by  tin  pipes  to  the  different  rooms  of  a  house.  Furnaces  have  been, 
of  late,  very  much  decried,  but  under  proper  regulation  they  are  very 
cheap,  economical,  and  even  healthful  means  of  ventilation  and  warming. 
The  heating  surface  should  be  very  large,  the  pot  thick,  or  even  encased 
with  fire-brick,  that  it  may  not  become  too  hot ;  there  should  be  a  plentiful 
supply  of  water  in  the  chamber  for  evaporation,  perhaps  also  beneath  the 
opening  of  each  register ;  the  air  supply  should  always  be  drawn  from  the 
outer  air,  and  unobjectionable  sources,  through  ample  and  tight  ducts, 
without  any  chance  of  draft  from  the  cellar;  the  pot,  and  all  joints  in 


304:  ARCHITECTURAL   DRAWING. 

the  radiator,  should  be  perfectly  gas-tight,  so  that  nothing  may  escape 
from  the  combustion  into  the  air-chamber.  With  these  provisions  on  a 
sufficient  scale,  and  proper  means  for  distribution  of  the  heated  air  and 
escape  of  foul  air,  almost  any  edifice  may  be  very  well  heated  and  ventilated. 
The  air  should  be  delivered  through  the  floor  or  the  base-board  of  the 
room,  and  at  the  opposite  side  from  the  flue  for  the  escape  of  foul  air, 
making  as  thorough  a  current  as  possible  across  the  room,  and  putting  the 
whole  air  in  motion.  In  dwelling-houses  the  fireplace  will  serve  the  best 
means  of  exit ;  in  public  rooms  distinct  flues  will  have  to  be  made  for  this 
purpose,  and  they  should  be  of  ample  dimensions,  with  openings  at  the 
floor,  and  means  should  be  provided  for  heating  this  flue.  An  architect,  in 
laying  out  flues  for  heating  and  ventilation,  should,  both  in  plan  and  ele- 
vation, fix  the  position  of  hot  and  foul  air  flues,  and  trace  in  the  current 
of  air,  always  keeping  in  mind  that  the  tendency  of  hot  air  is  to  rise ;  he 
will  then  see  that  if  the  exit-opening  be  directly  above  the  entrance-flue, 
the  hot  air  will  pass  out,  warming  the  room  but  little ;  if  the  exit-opening 
be  across  the  room  and  near  the  ceiling,  the  current  will  be  diagonal, 
with  a  cold  corner  beneath,  where  there  will  be  very  little  circulation  or 
warmth.  To  heat  the  exit-flue,  a  very  simple  way  is,  to  make  the  furnace- 
flue  of  iron,  and  let  it  pass  up  centrally  through  the  exit-flue. 

Size  for  furnace  and  ducts,  as  given  by  an  old  established  maker: 
Largest  pot,  22"  diam. ;  air-chamber,  4'  x  4'  4"  x  5'  6" ;  air-box  or  cold-air 
duct,  3  sq.  ft.  of  sectional  area,  with  a  slide  damper ;  hot-air  duct,  main 
riser,  say  22"  x  12",  or  equal  area,  branches  according  to  the  size  of  room 
and  position ;  water-trough  in  air-chamber,  about  3  cub.  ft.  capacity.  This 
furnace  would  be  adequate  to  heat  one  of  our  city  houses  of  25  ft.  fron.t  x 
60  ft.  deep,  with  registers  as  high  as  the  third  story.  Furnaces  vary  in  the 
size  of  the  pot  to  as  small  as  16"  diam.,  and  the  accessories  may  be  reduced 
proportionately. 

.  Steam  and  hoi-water  circulation  are  applied  to  the  heating  of  buildings 
by  means  of  wrought  or  cast  iron  pipes  connected  with  boilers.  In  the 
simplest  form,  as  common  in  workshops  and  factories,  steam  is  made  to  give 
warmth  without  ventilation  by  direct  radiation  from  wrought-iron  pipes, 
with  or  without  coils  or  radiators.  The  general  arrangement  is  by  rows  of 
£"  or  V  pipe  hung  against  the  walls  of  the  room  as  close  to  the  base  as 
possible,  one  foot  of  f "  pipe  being  considered  adequate  to  heat  50  cub.  ft. 
of  space  ;  if  there  are  many  windows  in  the  room,  or  the  building  is  very 
much  exposed,  more  length  should  be  allowed.  Steam  is  used  at  either 
low  or  high  pressures,  exhaust  or  live  steam ;  the  rising  main  should  be  as 
near  to  boiler  or  engine  as  possible,  and  the  heating  pipes  should  have  a 


ARCHITECTURAL   DRAWING.  305 

drip  or  descent  of  about  •£"  in  10  ft.,  away  from  rising  main,  and  the  con- 
densed water  should  be  returned  directly  or  indirectly  to  the  boiler. 

"When  ventilation  is  combined  with  steam  or  hot- water  heating,  coils  or 
radiators  are  placed  in  air-chambers,  through  which  the  outer  air  is  intro- 
duced and  distributed  by  ducts  throughout  the  building,  like  a  hot-air 
furnace,  and  a  great  improvement  over  this  is  that  the  metallic  surfaces 
brought  in  contact  with  the  air  in  the  first  case  usually  range  from  200°  to 
250°,  whilst  the  pot  of  the  air-furnace  is  often  near  a  white  heat.  In  a 
sanitary  point  of  view,  hot-water  or  low-steam  coils  in  air-chambers  are  by 
far  the  healthiest  means  of  warming  and  ventilation ;  the  greatest  objec- 
tion is  their  expense,  the  care  requisite  in  attending  them,  the  danger  of 
freezing  and  bursting  the  pipes  if  worked  intermittently  in  winter.  In  the 
arrangement  it  is  usual,  in  dwelling-houses,  to  place  the  coils  at  different 
points  in  the  cellar,  as  near  as  possible  beneath  the  rooms  to  be  heated. 
In  public  buildings  frequently  a  very  large  space  in  the  cellar  is  occupied 
by  the  coils,  into  which  the  air  is  forced  by  a  fan,  and  then  distributed  by 
flues  or  ducts  throughout  the  building.  Hot-water  circulation  is  very 
much  used  in  the  heating  of  forcing  or  hot-houses,  as  the  mass  of  water 
once  heated  acts  as  a  regulator,  and  there  are  less  liabilities  to  change  of 
temperature  by  inattention  to  the  fires. 

All  inlet  or  outlet  ventilating  flues  should  be  provided  with  dampers 
or  registers,  to  control  the  supply  or  discharge  of  air,  cutting  it  off  when 
sufficient  heat  is  secured,  or  retaining  the  warmth  when  ventilation  is  not 
required. 

Much  has  been  written  on  the  subject  of  ventilation  and  warming,  and 
many  expedients,  undoubtedly  adequate  in  themselves,  have  failed,  from 
the  carelessness  of  servants  and*  from  want  of  attention.  The  grand  requi- 
site seems  to  be,  something  that  will  be  sure,  and  will  not  get  out  of  order. 
It  is  now  but  a  half  a  century  since  gas  was  introduced  for  lighting;  it  has 
been  applied  for  cooking  and  warming,  but  not  to  a  large  extent  eco- 
nomically ;  whether  it  may  be  brought  into  general  use  for  this  purpose 
is  a  problem  yet  to  be  solved ;  but  steam,  as  now  applied  in  most  New 
England  factories,  from  a  central  set  of  boilers,  could  easily  be  applied  to 
the  warming  and  ventilation  of  many  houses,  and  for  many  culinary  pur- 
poses, and  with  gas  would  supply  all  requirements. 

Ventilators. — Patented  expedients  are  numerous  to  excite  drafts  in  flues 
or  ducts  that  will  not  draw,  some  movable  and  some  immovable,  and  are  to 
be  purchased  at  almost  every  tinsmith's.  The  simplest  form  is  that  in  which 
the  wind,  blowing  across  the  top  of  the  chimney,  is  given  such  a  direction 

that  it  creates  a  vacuum  in  the  flue  beneath ;  but  when  adequate  heat  can 
20 


306  AECHITECTUEAL   DBA  WING. 

be  applied,  either  by  a  jet  of  burning  gas,  by  steam  coil,  or  direct  fire,  a 
circulation  can  be  assured.  In  many  large  public  institutions,  charitable, 
educational,  and  legislative,  forced  circulation  by  means  of  fans  is  very 
common ;  in  most  cases,  by  drawing  the  air  from  the  outside  of  the  build- 
ing, and  forcing  it  under  pressure  into  the  interior,  the  escape  being  out- 
ward through  all  flues  and  cracks,  or  ventilation  by  &  plenum,  as  it  is  called ; 
in  other  cases  the  fan  drawing  its  air  directly  from  the  interior,  and 
reducing  the  pressure,  to  overcome  which  the  air  rushes  in  from  the  out- 
side through  the  hot-air  ducts  and  cracks,  or  ventilation  by  vacuum. 

Ventilation  has  been  treated  as  applicable  to  dwellings  or  buildings  for 
the  occupancy  of  mankind,  but  the  architect  must  understand  that  fresh  air 
is  equally  necessary  for  animals  and  even  plants.  In  stables  there  must  be 
provision  for-  the  escape  of  foul  air  and  the  introduction  of  fresh  air,  with- 
out injurious  drafts  or  too  great  reduction  of  temperature.  Care  should 
also  be  taken,  in  the  discharge  of  the  foul  air,  that  it  does  not  escape  into 
the  hay-loft,  to  be  condensed  on  the  fodder,  to  make  it  unpalatable.  Barn 
and  stable  should  be  under  separate  roofs ;  the  barn  may  be  ventilated  by 
a  cupola,  with  blinds,  on  the  roof;  but  if  the  stable  can  be  ventilated  some- 
what similar  to  the  room  of  a  dwelling-house,  by  taking  in  air  at  one  side 
and  letting  it  out  at  the  other,  so  much  the  better ;  and  as  the  animals 
cannot  have  a  radiant  fire  to  warm  themselves,  give  them  sunlight. 

Drainage,  in  a  sanitary  point  of  view,  is  almost  as  necessary  as  venti- 
lation ;  as  applied  to  agriculture  by  means  of  subsoil  tile  drains,  it  increases 
the  crops,  and  conduces  to  the  healthfulness  of  the  country ;  to  dwellings, 
through  sewers  and  drains,  it  removes  one  of  the  most  prolific  causes  of 
disease.  If  the  land  be  wet  and  springy,  some  provision  for  drainage  must  be 
made  below  the  level  of  the  cellar.  A  simple  and  effectual  way  is  by  enclos- 
ing the  foundation,  as  far  as  may  be  necessary,  by  drains,  and  leading  the 
water  off;  if  the  cellar  walls  be  laid  in  concrete  and  bottom  cemented,  the 
precaution  will  be  sufficient.  But  the  foul  water  or  house  waste  is  also  to 
be  disposed  of ;  in  cities  this  is  done  by  connection  with  street  sewers,  which 
are  usually  placed  deep  enough  to  drain  the  cellar.  The  sewer  pipes  or 
drain  connections  with  street  sewers  are,  in  this  city,  established  at  6" 
diam.,  usually  made  of  vitrified  stone-ware  pipe,  and  the  pitch  or  descent 
should  be  as  uniform  as  possible,  and  at  least  ^'  in  10  ft.  Many  make 
use  of  cement  pipe  instead  of  stone-ware,  as  somewhat  cheaper.  Iron 
pipe  is  used,  of  from  5"  to  6"  diam.,  when  the  pipe  cannot  be  covered.  In 
cities  the  sewer  pipe  is  used  for  the  discharge  of  the  rain  water  from  the 
leaders,  but  in  country  residences  this  is  usually  collected  in  cisterns  for 
washing  purposes ;  the  sewer  pipes,  being  merely  for  the  discharge  of  the 


ARCHITECTURAL   DRAWING. 


307 


foul  water  of  the  house,  are  usually  made  about  4?  diam.  The  purpose  of 
all  house  apparatus  of  water-closets,  sinks,  waste  and  sewer  pipes,  is  for 
the  immediate  and  speedy  conveyance  of  every  particle  thrown  into  them 
out  of  the  house  into  the  final  collecting  place,  and  that  there  should  be  no 
smell  escape  from  them  into  the  interior  of  the  house  or  by  return.  For 
the  first  the  pipes  should  be  laid  with  as  few  bends  as  possibje,,  and  with  a 
good  descent;  for  the  latter  every  waste  pipe  should  be  trapped.  The 
most  efficient  traps  are  water  closures,  which  admit  the  passage  of  water, 
but  not  the  return  of  air.  Thus,  in  Fig.  125,  which  is  a  section  of  a  form 


Fig.  125. 


Fig.  126. 


of  privy  vault,  recommended  by  the  Board  of  Health  for  city  tenement 
houses,  the  trap  is  made  by  the  bending  down  of  the  sewer  pipe  S  into  the 
surface  of  the  vault,  or  a  sipfion  trap;  and  in  fig.  126  by  the  wall  W 
extending  below  the  level  of  the  bottom  of  the  sewer  pipe,  corresponding 
to  the  catch-basin  trap ;  the  same  purpose  is  served  in  a  waste  pipe  by 
making  a  bend  or  S  trap  in  it.  The  pipes  L  L  are  waste  pipes  from  house 
or  roof  leaders  to  flush  the  vaults. 

The  final  receptacle  of  waste  for  country  houses  is  one  of  the  most  diffi- 
cult problems  to  be  solved  by  the  architect.  If  the  premises  are  extensive, . 
it  may  be  made  at  any  place  on  a  descending  slope  from  the  house,  the 
walls  laid  in  stone  dry,  so  that  the  earth  may  absorb  the  liquid  portion. 
In  this  case  it  should  be  distant  from  the  house  and  the  water  supply,  so 
that  there  may  be  no  offence  from  the  smell,  nor  any  contamination  from 
infiltration ;  a  sink  of  this  kind,  even  in  the  most  porous  earth,  becomes, 
in  a  few  years,  clogged,  and  another  has  to  be  dug  in  fresh  earth.  The 
size  of  the  sink  will,  of  course,  depend  on  the  character  of  the  soil  and  the 


308  ABCHTTECTUBAL   DRAWING. 

quantity  of  waste ;  the  top  of  some  are  domed  in  brick  and  cement,  but 
the  cheapest  covering,  which  will  endure  as  long  as  the  sink  continues  to 
perform  its  functions  without  overflow,  is  locust  sleepers  and  flag  covers, 
with  1  ft.  to  18"  of  earth  above.  In  many  cases  a  tight  cesspool  becomes 
necessary  to  act  as  a  sort  of  settling  reservoir,  whilst  the  liquid  portions  are 
permitted  to  escape  through  an  overflow  pipe ;  but  this  only  mitigates  the 
evil,  there  is  still  the  difficulty  where  to  lead  the  overflow.  In  some  few 
instances  the  waste  is  pumped  out  of  the  cesspool  and  distributed  as  manure 
over  the  land,  no  liquid  being  permitted  to  escape ;  where  this  is  done 
daily  there  is  no  cause  of  offence,  and  the  return  is  in  increased  vegetation. 

Lighting. — Make  such  provisions  and  arrangements  of  rooms  and  pas- 
sages that  they  may  secure  as  much  as  possible  of  their  light  from  the 
natural  source,  and  direct  without  the  aid  of  reflectors.  Sunlight  in  the 
dwelling  conduces,  with  fresh  air  and  good  drainage,  to  health  and  happi- 
ness. Let  there  be  no  dark  nooks  and  corners  in  which  artificial  light  is 
necessary  in  the  daytime.  Make  windows  as  required,  with  movable  sash 
for  ventilation,  and  blinds  for  regulating  the  light. 

Water  Supply. — In  all  dwellings  there  must  be  some  source  of  water 
supply,  by  springs,  wells,  cisterns,  or  aqueducts.  Most  of  our  cities  have 
sufficient  water  works,  and  in  that  case  connection  therewith  to  the  interior 
of  houses  is  almost  compulsory.  In  country  houses,  on  the  other  hand,  the 
cases  are  very  few  in  which  water  can  be  supplied  directly  by  gravitation 
to  the  upper  stories.  Wells  and  pumps,  force  or  lift,  are  the  most  usual 
expedients,  and  when  well  water  is  hard,  the  soft  water  for  washing 
may  be  drawn  from  cisterns,  which  receive  the  water  from  the  roof.  The 
quantity  necessary  for  each  household  varies  with  the  wants  and  habits  of 
the  occupants;  from  7  to  10  gallons  can  be  reckoned  as  amply  sufficient  for 
each  person  if  used  legitimately.  In  the  construction  of  wells  to  supply 
a  house  by  gravitation,  it  is  a  very  good  plan  to  dig  horizontally  into  a  fit 
place  on. the  side-hill,  till  the  supply  running  doWn  the  trench  is  sufficient, 
then  make  a  stone  culvert  from  the  upper  end  down  as  far  as  necessary, 
closing  the  end  by  a  cement  wall,  for  a  reservoir,  and  covering  the  whole 
with  earth,  leaving  a  man-hole  for  access.  The  size  of  pipe  depends  upon 
the  head  above  the  outlets,  and  the  quantity  of  water  required ;  in  this 
city  the  tap  permitted  in  the  street  mains  for  a  dwelling-house  is  f  " ;  for 
country  houses  the  pipe  should  generally  not  be  less  than  1"  diam. 

Lead  pipes  are  almost  the  only  ones  used  in  plumbing,  but  care  should 
be  taken  in  testing  the  effect  of  water  to  be  used  on  lead,  as,  in  many  cases, 
it  acts  on  the  metal  and  becomes  a  most  subtle  poison.  In  such  cases  it  is 
better  to  use  pure  tin  pipes,  tin-cased  pipes,  or  galvanized  iron. 


ARCHITECTURAL   DRAWING.  309 

Principles  of  Design. — It  is  not  intended  in  this  book,  professedly 
treating  of  architecture  only  in  its  most  mechanical  phase  of  drawing,  to 
give  a  history  of  it  as  an  art,  or  a  treatise  on  the  distinctions  of  styles. 
To  one  anxious  to  acquire  knowledge  in  this  department  we  refer,  as  the 
very  best  compendium  within  our  knowledge,  to  Ferguson's  "  Hand-book 
of  Architecture."  The  study  of  this  work  will  give  direction  to  a 
person's  observation,  but,  without  referring  to  actual  examples,  mere  read- 
ing will  be  of  little  nse.  Drawings  give  general  ideas  of  the  character 
of  buildings,  but  no  idea  of  size  or  of  the  surroundings  of  a  building.  Many 
a  weak  design,  especially  in  cast-iron  buildings,  acquires  a  sort  of  strength 
by  the  number  of  its  repetitions,  giving  an  idea  of  extent ;  and  many  a 
beautiful  design  on  paper  has  failed  in  its  execution,  being  dwarfed  by  its 
surroundings.  With  regard  to  the  style  of  a  building,  there  are  none  of 
the  ancient  styles  in  their  purity  adapted  to  present  requirements ;  our 
churches  and  theatres  are  more  for  the  gratification  of  the  ear  than  the 
eye,  and  the  comforts  of  our  domestic  architecture,  and  the  requirements 
of  our  stores  and  warehouses,  are  almost  the  growth  of  the  present  cen- 
tury. For  a  design,  look  first  to  the  requirements  of  the  structure,  the 
purposes  to  which  it  is  to  be  applied ;  sketch  the  plan  first,  arrange  the  divi- 
sions of  rooms,  the  openings  for  doors  and  windows,  construct  the  sections, 
and  then  the  elevations  first  in  plain  outline  ;  modify  each  by  the  exigencies 
of  construction. 

"  Construction,  including  in  the  term  the  disposition  of  a  building  in 
reference  to  its  uses,  is  by  some  supposed  to  be  the  common  part  of  the  art 
of  architecture,  but  it  is  really  the  bone,  muscle,  and  nerve  of  architecture, 
and  the  arts  of  construction  are  those  to  which  the  true  architect  will  look, 
rather  than  to  rules  and  examples,  for  the  means  of  producing  two  at  least 
of  the  three  essential  conditions  of  building  well,  commodity r,  firmness,  and 
delight,  which  conditions  have  been  aptly  said  to  be  the  end  of  architecture 
as  of  all  creative  arts. 

"  The  two  great  principles  of  the  art  are :  First,  that  there  should  be  no 
features  about  a  building  which  are  not  necessary  for  convenience,  con- 
struction, or  propriety ;  second,  that  all  ornament  should  consist  of  en- 
richment of  the  essential  construction  of  the  building. 

"  The  neglect  of  these  two  rules  is  the  cause  of  all  the  bad  architecture 
of  the  present  time.  Architectural  features  are  continually  tacked  on  build- 
ings with  which  they  have  no  connection,  merely  for  the  sake  of  what  is 
termed  effect,  and  ornaments  are  continually  constructed  instead  of  forming 
the  decoration  of  construction  to  which  in  good  taste  they  should  always  be 
subservient.  The  taste  of  the  artist  ought  to  be  held  merely  ancillary  to 


310  ARCHITECTURAL   DRAWING. 

truthful  disposition  for  structure  and  service.  The  soundest  construction 
is  the  most  apt  in  the  production  or  the  reproduction,  it  may  be,  of  real 
art.  The  Eddystone  lighthouse  is  well  adapted  to  its  uses ;  it  is  commo- 
dious, firm  and  stable  almost  to  a  miracle,  and  its  form  is  as  beautiful  in 
outline  to  the  delight  of  the  eye,  as  it  is  well  adapted  to  break  and  miti- 
gate the  force  of  the  sea  in  defence  of  its  own  structure.  The  Great  Exhi- 
bition Building  of  1851  was  most  commodious  for  the  purposes  of  an  ex- 
hibition, firm  enough  for  the  temporary  purpose  required  of  it,  and  there 
was  delight  in  the  simplicity  and  truth  of  its  combinations ;  and  all  this 
may  be  said  to  have  grown  out  of  propriety  of  construction,  as  applied  to 
the  material,  cast  iron.  The  use  of  unfitting  material,  or  fitting  material, 
inappropriately,  leads  almost  entirely  to  incommodiousness,  infirmity,  and 
offence,  or  some  of  them. 

"  Out  of  truth  in  structure,  and  that  structure  of  a  very  inartificial  sort, 
grow  the  beautiful  forms  of  the  admirable  proportions  found  in  the  works 
of  the  Greeks ;  and  out  of  truth  in  structure  with  the  strictest  regard  to 
the  necessities  of  the  composition  and  of  the  material  employed,  and  that 
structure  as  full  of  artifice,  as  the  artifice  employed  is  of  truth  and  simpli- 
city, grew  the  classical  works  vulgarly  called  Gothic,  but  now  character- 
istically designated  as  Pointed,  from  the  arch  which  is  the  basis  of  the  style. 
Structural  untruth  is  not  to  be  justified  by  authority ;  neither  Sir  Chris- 
topher Wren,  nor  the  Athenian  exemplars  of  Doric  or  Ionic  in  the  Propy- 
laeum  and  in  the  Minerva  Polias,  with  their  irregular  and  inordinately  wide 
intercolumniation,  can  persuade  even  the  untutored  eye  to  accept  weak- 
ness for  strength,  or  what  is  false  for  truth. 

"  The  Greek  examples  offer  the  most  beautiful  forms  for  mouldings,  and 
the^  Grecian  mode  of  enriching  them  is  unsurpassed.  It  should  be  borne 
in  mind  that  the  object  in  architectural  enrichment,  is  not  to  show  orna- 
ment, but  to  enrich  the  surface  by  producing  an  effective  and  pleasing 
variety  of  light  and  shade ;  but  still,  although  ornament  should  be  a  second- 
ary consideration,  it  will  develop  itself,  and  therefore  should  be  of  elegant 
form  and  composition." 

"We  have  quoted  thus  at  some  length  from  the  article  Architecture, 
Encyclopaedia  Britannica,  because  with  many  authority  is  necessary,  and 
they  distrust  their  own  powers  of  observation  and  analysis ;  all  must  feel 
the  truth  of  the  above,  but  in  practice  it  is  very  little  appreciated  or  carried 
out.  The  present  taste  in  architecture,  as  in  the  theatre,  is  for  the  spec- 
tacular ;  breadth  or  dignity  of  effect  is  not  popular ;  edifices  are  not  only 
covered  with,  but  built  up  in  ornament ;  and  construction  is  but  secondary. 
The  French,  having  a  building  stone  that  is  very  easily  worked,  cut  merely 


ARCHITECTURAL   DRAWING.  311 

the  joints,  leaving  the  rough  outer  surface  to  be  worked  after  it  is  laid ; 
chopping  out  mouldings  and  ornaments  almost  as  readily  as  though  it  were 
in  plaster,  and  the  surface  when  finished  is  covered  with  enrichments  in 
low-relief.  The  fashion  thus  set  is  imitated  in  this  country  at  immense 
cost,  in  the  most  unfitting  materials — marble  and  granite.  Our  architec- 
tural buildings  express  fitly  our  condition — a  rich  country,  recent  and  easily- 
acquired  wealth,  and  a  desire  and  rivalry  ,to  exhibit  it,  or  a  display  as  a 
means  of  advertising,  and  in  this. truth  of  expression  will  have  an  archaeo- 
logical interest ;  although  it  does  not  contribute  much  to  present  excellence 
in  construction,  it  still  has  this  value,  that  the  architect  or  constructor  need 
be  governed  by  no  rules  or  principles,  he  can  make  experiments  on  a 
pretty  extensive  scale,  and  out  of  much  bad  construction  even  forms  and 
ornament  may  spring  up  which  will  stand  the  test  of  time,  and  form  a 
nucleus  of  a  new  style  adapted  to  the  present  wants. 

A  new  building  material  for  building  purposes — cast  iron — has  recently 
come  into  use,  but,  with  the  exception  of  exhibition-buildings  before 
mentioned,  has  not  yet  been  treated  distinctively;  buildings  erected  with 
it  have  been  copies  of  those  in  stone,  and  have  been  even  imitated  in 
color.  For  the  first  story  of  stores,  where  space  is  necessary  for  light  and 
the  exhibition  of  wares,  cast-iron  columns  are  almost  invariably  used, 
but  are  objected  to  architecturally,  that  they  look  too  weak  for  the  sup- 
port of  the  piles  of  brick  and  stone  above  them.  The  objection  should  not 
be  to  the  use,  but  that  the  truth  of  the  adequate  strength  of  the  cast  iron 
is  not  conveyed  by  the  form  or  color.  No  one  objects  that  the  ancles  of 
Atlas  look  too  light  to  support  the  massive  figure  and  globe,  or  wishes  it 
seated  to  give  the  idea  of  stability ;  so  if  the  columns  and  lintels  were  some 
other  form,  than  Greek  or  Komari  with  immense  intercolumniations,  and 
colored  fitly,  the  appearance  of  weakness  would  be  entirly  lost  sight  of. 

In  conclusion.  Some  years  since  the  author  made  an  experiment  on 
building  according  to  the  principles  above  laid  down,  construction  and 
adaptation  and  expression  of  purpose.  As  regards  convenience  and 
strength,  it  was  found,  on  occupation,  all  that  could  be  wished ;  in  regard 
to  the  third  requirement,  delight,  a  sketch  is  given  on  page  312,  and  is 
open  to  the  reader's  criticism.  Some  allowance  should  be  made  for  absence 
of  color  in  the  sketch,  which  contributed  much  to  architectural  effect. 
Posts,  lintels,  window-frames  and  sash,  and  ornamental  letters,  were  of  iron, 
and  painted  a  very  deep  green;  the  structure  was  of  brick,  with  sills  and 
bands  of  rubbed  Ulster  Milestone,  roof  of  "Welsh  slate.  The  building 
occupied  one  corner  of  Greene  and  Houston  Streets,  in  this  city,  but  was 
burned,  and  cannot,  therefore,  be  referred  to  practically. 


312 


ARCHITECTURAL   DRAWING. 


SHADING    AND    SHADOWS.  313 


SHADING  AND   SHADOWS. 

LIGHT  is  diffused  through  space  in  straight  lines,  and  the  lines  of  light 
are  called  rays.  When  the  source  of  light  is  situated  at  a  very  great  dis- 
tance from  the  illuminated  objects,  as  in  the  case  of  the  sun  with  relation 
to  the  earth,  the  rays  of  light  do  not  sensibly  diverge,  and  may  be  regarded 
as  exactly  parallel  to  each  other.  Such  is  the  case  in  mechanical  draw- 
ings, where  the  objects  to  be  represented  are  always  regarded  as  illumi- 
nated by  the  solar  light. 

Light  is  called  direct  when  it  is  transmitted  to  an  object  without  the 
intervention  of  any  opposing  medium.  But  as  all  bodies  subjected  to  the 
action  of  light  possess,  in  a  greater  or  less  degree,  the  property  of  giving 
out  a  certain  portion  of  it  to  the  surrounding  objects,  this  reflected  light 
becomes  in  its  turn,  though  with  greatly  diminished  intensity,  a  source  of 
illumination  to  those  objects  which  are  deprived  of  direct  light. 

Everything  which  tends  to  intercept  or  prevent  the  direct  light  from 
falling  in  upon  a  body,  produces  upon  the  surface  of  that  body  a  degree 
of  obscurity  of  greater  or  less  intensity  ;  this  is  called  a  shade  or  shadow. 
Such  effects  are  usually  classified  as  direct  shadows  and  cast  shadows. 

The  shade  proper,  or  direct  shadow,  is  that  which  occurs  on  that  por- 
tion of  the  surface  of  a  body  which  is  situated  opposite  to  the  enlightened 
part,  and  is  the  natural  result  of  the  form  of  the  body  itself,  and  of  its  posi- 
tion with  regard  to  the  rays  of  light.  The  cast  shadow,  on  the  other  hand, 
is  that  which  is  produced  upon  the  surface  of  one  body  by  the  interposi- 
tion of  another  between  the  former  and  the  source  of  light ;  thus  inter- 
cepting the  rays  which  would  otherwise  illuminate  that  surface.  An 
illustration  of  this  distinction  is  afforded  in  the  pyramid  represented  at 
fig.  1,  Plate  LXXX.,  where  the  shade  proper  is  shown  upon  that  half  of 
the  figure  which  is  denoted  by  the  letters  D'  E'  G'  F'  in  the  plan,  while  the 
cast  shadow  occupies  the  space  comprised  between  the  lines  E7  e  and  F;  d 
on  the  horizontal  plane  of  projection.  Cast  shadows  may  also  obviously 


314:  SHADING   AND   SHADOWS. 

be  produced  upon  the  surface  of  a  body  by  the  form  of  the  body  itself ;  as, 
for  example,  if  it  contain  projecting  or  concave  parts. 

The  limit  of  the  direct  shadow  in  any  body,  whatever  may  be  its  form 
or  position,  is  a  line  of  greater  or  less  distinctness,  termed  the  line  of  sep- 
aration between  light  and  shade',  or,  more  shortly,  the  line  of  shade  •  this 
line  is,  of  course,  determined  by  the  contact  of  the  luminous  rays  with  the 
surface  of  the  body ;  and  if  these  rays  be  prolonged  till  they  meet  a  given 
surface,  by  joining  all  the  points  of  intersection  with  that  surface,  we  ob- 
tain the  outline  of  the  shadow  cast  upon  it  by  the  part  of  the  body  which 
is  deprived  of  light. 

The  rays  of  light  being  regarded  as  parallel  to  each  other,  it  is  obvious 
that  in  the  delineation  of  shadows,  'it  is  only  necessary  to  know  the  direc- 
tion of  one  of  them  ;  and  as  that  direction  is  arbitrary,  we  have  adopted 
the  usual  and  confessedly  the  most  convenient  mode  of  regarding  the  ravs 
as  in  all  cases  falling  in  the  direction  of  the  diagonal  of  a  cube,  of  which 
the  sides  are  parallel  to  the  planes  of  projection.  The  diagonal  in  projec- 
tion upon  the  vertical  and  horizontal  planes  lies  at  an  angle  of  45°  with 
the  ground-line  ;  and  thus  the  light  in  both  elevation  and  plan  appears  at 
the  angle  of  45°.  In  illustration,  let  E,  E'  (fig.  1,  pi.  LXXIX.)  be  the  pro- 
jections of  a  ray  of  light  in  elevation  and  plan ;  and  let  A,  A',  those  of  a 
point  of  which  the  shadows  are  required  to  be  projected  upon  the  vertical 
plane  X  Y.  Draw  the  straight  lines  A  a,  A'  a',  parallel  to  the  lines  E,  E', 
and  from  a' ',  where  the  line  A'  a'  meets  the  plane  X  T,  draw  the  perpen- 
dicular a'  a  to  meet  the  oblique  line  A  a;  then  the  intersection  a  is  the 
position  of  the  shadow  of  the  point  A. 

In  the  following  illustrations,  the  same  letter  accented,  is  employed  in 
the  plan  as  in  the  elevation,  to  refer  to  the  same  point  or  object. 

The  projections  of  the  diagonals  of  the  imaginary  cube  which  denote 
the  direction  of  the  rays  of  light  being  equal  in  both  planes,  it  follows 
that  in  all  cases,  and  whatever  may  be  the  form  of  the  surface  upon  which 
the  shadow  is  cast,  the  oblique  lines  joining  the  projections  of  the  point 
which  throws  the  shadow,  and  that  which  denotes  it,  are  also  equal.  Thus 
the  line  A  a  in  the  elevation  is  equal  to  the  line  A!  a'  in  the  plan.  Hence 
it  will  in  some  cases  be  found  more  convenient  to  use  the  compasses 
instead  of  a  geometrical  construction;  as,  for  example,  in  place  of  project- 
ing the  point  a'  by  a  perpendicular  to  the  ground-line,  in  order  to  obtain 
the  position  of  the  required  shadow  «,  that  point  may  be  found  by  pimply 
setting  off  upon  the  line  A  a  a  distance  equal  to  A!  a'. 

Plate  LXXIX.,  fig.  1. — Required  to  determine  the  shadow  cast  upon 
the  vertical  wall  X  Y  ~by  the  straight  line  A  B. 


SHADING  AND  SHADOWS.  315 

It  is  obvious  that  in  tins  case  the  shadow  itself  will  be  a  straight  line ; 
hence,  to  solve  the  problem,  it  is  only  necessary  to  find  two  points  in  that 
line.  We  have  seen  that  the  position  of  the  shadow  thrown  by  the  point 
A  is  at  a;  by  a  similar  process  we  can  easily  determine  the  point  b,  the 
position  of  shadow  thrown  by  the  opposite  extremity  B  of  the  given  line ; 
the  straight  line  a  b,  which  joins  these  two  points,  is  the  shadow  required. 

It  is  evident  from  the  construction  of  this  figure,  that  the  line  a  l>  is 
equal  and  parallel  to  the  given  line  A  B ;  this  results  from  the  circum- 
stance that  the  latter  is  parallel  to  the  vertical  plane  X  Y.  Hence,  when 
a  line  is  parallel  to  a  plane,  its  shadow  upon  that  plane  is  a  line  which  is 
equal  and  parallel  to  it. 

Suppose  now  that,  instead  of  a  mere  line,  a  parallel  slip  of  wood  or 
paper,  A  B  C  D,  be  taken,  which,  for  the  sake  of  greater  simplicity,  we 
shall  conceive  as  having  no  thickness.  The  shadow  cast  by  this  object 
upon  the  same  vertical  plane  X  Y  is  a  rectangle  a  1  c  d,  equal  to  that 
which  represents  the  projection  of  the  slip,  because  all  the  edges  of  the 
latter  are  parallel  to  the  plane  upon  which  the  shadow  is  thrown.  Hence, 
in  general,  when  any  surface,  whatever  may  be  its  form,  is  parallel  to  a 
plane,  its  shadow  tliroivn  upon  that  plane  is  a  figure  similar  to  it,  and  simi- 
larly situated.  This  principle  facilitates  the  delineation  of  shadows  in 
many  cases.  In  the  present  example,  an  idea  may  be  formed  of  its 
utility  ;  for,  after  having  determined  the  position  of  any  one  of  the  points 
a,  o,  c,  d,  the  figure  may  be  completed  by  drawing  lines  e'qual  and  parallel 
to  the  sides  of  the  slip,  without  requiring  to  go  through  the  operations  in 
detail. 

Fig.  2. — "When  the  object  is  not  parallel  to  the  given  plane,  the  cast 
shadow  is  no  longer  a  figure  equal  and  similarly  placed  ;  the  method  of 
determining  it  remains,  however,  unchanged ;  thus,  take  the  portion  A  E 
of  the  slip  A  B,  which  throws  its  shadow  on  the  plane  X  Y ;  draw  the 
lines'  A  a,  E  e,  C  c,  F/,  and  A'  a',  E'  e',  parallel  to  the  rays  of  light ;  make 
A  a  and  C  c  equal  to  A!  a'',  and  E  e  and  F/  equal  to  E'  e' \  connect  a  ef"c, 
and  we  have  the  outline  of  the  shadow  of  the  slip  A  E. 

By  an  exactly  similar  construction  we  have  the  shadow  of  the  portion 
E  B  on  the  plane  Y  Z,  which  being  inclined  to  the  plane  of  projection  in 
a  direction  contrary  to  X  Y,  necessarily  causes  the  shadow  to  be  broken, 
and  the  part  e  d  to  lie  in  a  contrary  direction  to  af. 

Fig.  3  still  further  illustrates  the  determination  of  the  shadow  of  the 
slip  upon  a  moulding  placed  on  the  plane  X  Y  parallel  to  the  slip. 

Fig.  4. — To  find  the  shadow  cast  by  a  straight  line  A  B  upon  a  curved 
surface,  either  convex  or  concave,  whose  horizontal  projection  is  repre- 
sented by  the  line  X  e'  Y. 


316  SHADING   AND    SHADOWS. 

It  has  been  already  explained,  that  the  shadow  of  a  point  upon  any 
surface  whatever  is  found  by  drawing  a  straight  line  through  that  point, 
parallel  to  the  direction  of  the  light,  and  marking  its  intersection  with  the 
given  surface.  Therefore,  through  the  projections  A  and  A'  of  one  of  the 
points  in  the  given  straight  line,  draw  the  lines  A  a,  A'  a',  at  an  angle  of 
45°;  and  through  the  point «',  where  the  latter  meets  the  projection  of  the 
given  surface,  raise  a  perpendicular  to  the  ground-line  ;  its  intersection 
with  the  line  A  a  is  the  position  of  the  shadow  of  the  first  point  taken  ; 
and  so  for  all  the  reversing  points  in  the  line. 

If  it  be  required  to  delineate  the  entire  shadow  cast  by  a  slip  A  B  C  D, 
by  the  construction  above  explained,  trace  two  equal  and  parallel  curves 
aeb,  cfd,  which  will  represent  the  shadows  of  the  sides  A  B  and  C  D  ; 
while  those  of  the  remaining  sides  will  be  found  denoted  by  the  vertical 
straight  lines  a  c  and  o  d,  also  equal  and  parallel  to  each  other,  and  to  the 
corresponding  sides  of  the  figure,  seeing  that  these  are  themselves  vertical 
and  parallel  to  the  given  surfaces. 

Fig.  5. — When  the  slip  is  placed  perpendicularly  to  a  given  plane  X  Y, 
on  which  a  projecting  moulding,  of  any  form  whatever,  is  situated,  the 
shadow  of  the  upper  side  A'  B',  which  is  projected  vertically  in  A,  will  be 
simply  a  line  A  a  at  an  angle  of  45°,  traversing  the  entire  surface  of  the 
moulding,  and  prolonged  unbroken  beyond  it.  This  may  easily  be  de- 
monstrated by  finding  the  position  of  the  shadow  of  any  number  of  points 
such  as  D',  taken  at  pleasure  upon  the  straight  line  A'  B'.  The  shadow 
of  the  opposite  side,  projected  in  C,  will  follow  the  same  rule,  and  be  de^ 
noted  by  the  line  C  <?,  parallel  to  the  former.  Hence,  as  a  useful  general 
rule  :  in  all  cases  where  a  straight  line  is  perpendicular  to  a  plane  of  pro- 
jection, it  throws  a  shadow  upon  that  plane  in  a  straight  line,  forming  an 
angle  of  45°  with  the  ground-line. 

Fig.  6  represents  still  another  example  of  the  shadow  cast  by  the  slip 
in  a  new  position  ;  here  it  is  supposed  to  be  set  horizontally  in  reference 
to  its  own  surface,  and  perpendicularly  to  the  given  plane  X  Y.  Here 
the  shadow  commences  from  the  side  D  B,.  which  is  in  contact  with  this 
plane,  and  terminates  in  the  horizontal  line  a  c,  which  corresponds  to  the 
opposite  side  A  C  of  the  slip. 

Plate  LXXIX.,  fig.  7. — Required  to  find  the  shadow  cast  upon  a  verti- 
cal plane  X  Y  by  a  given  circle  parallel  to  it. 

Let  C,  C',  be  the  projections  of  the  centre  of  the  circle,  and  E,  B',  those 
of  the  rays  of  light. 

It  has  been  already  shown,  that  when  a  figure  is  parallel  to  a  plane, 
its  shadow  cast  upon  that  plane  is  a  figure  in  every  respect  equal  to,  and 


SHADING  AND  SHADOWS.  317 

symmetrical  with  it ;  therefore  the  shadow  cast  by  the  circle  now  under 
consideration  will  be  expressed  by  another  circle  of  equal  radius  ;  conse- 
quently, if  the  position  of  the  centre  of  this  new  circle  be  determined,  the 
problem  will  be  solved.  Now  the  position  of  the  shadow  of  the  central 
point  C,  according  to  the  rules  already  fully  developed,  is  easily  fixed  at 
c  j  from  which  point,  if  a  circle  equal  to  the  given  circle  be  described,  it 
will  represent  the  outline  of  the  required  shadow. 

Fig.  8. — "Wlien  the  circle  is  perpendicular  to  both  planes  of  projection, 
its  projection  upon  each  will  obviously  be  represented  by  the  equal  diam- 
eters A  B  and  C'  D',  both  perpendicular  to  the  ground-line.  In  this  case, 
to  determine  the  cast  shadow,  describe  the  given  circle  upon  both  planes, 
as  indicated  by  the  figures,  and  divide  the  circumference  of  each  into  any 
number  of  equal  parts  ;  then,  having  projected  the  points  of  division,  as 
A2,  C2,  E2,  &c.,  to  their  respective  diameters  A  B  and  C'D',  draw  from 
them  lines  parallel  to  the  rays  of  light,  which,  by  their  intersection  with 
the  given  plane,  will  indicate  so  many  points  in  the  outline  of  the  cast 
shadow. 

Fig.  11. — If  the  given  circle  be  horizontal,  its  shadow  cast  upon  the  ver- 
tical plane  X  Y  becomes  an  ellipse  which  must  be  constructed  by  means 
of  points,  as  indicated  by  the  figures  referred  to  above ;  that  is  to  say,  that 
in  the  circumference  of  the  circle  a  certain  number  of  points  are  to  be 
taken, -such  as  A'  D'  B',  &c.,  which  are  to  be  projected  successively  to 
A,  D,  B,  on  the  line  A  B,  and  through  each  of  these  points  lines  are  to  be 
drawn  parallel  to  the  direction  of  the  rays  of  light,  and  their  intersection 
with  the  given  plane  determined.  The  junction  of  all  these  points  will 
give  the  ellipse  a  d  &,  which  is  the  contour  of  the  required  shadow. 

Fig.  9  represents  a  circle  whose  plane  is  situated  perpendicularly  to 
the  direction  of  the  luminous  rays.  In  this  example  the  method  of  con- 
structing the  cast  shadow  does  not  differ  from  that  pointed  out  in  reference 
to  fig.  11,  provided  that  both  projections  are  made  use  of.  But  it  is  obvious, 
that  instead  of  laying  down  the  entire  horizontal  projection  of  this  circle, 
all  that  is  necessary  is  to  set  off  the  diameter  D'  E'  equal  to  A  B,  because 
the  shadow  of  this  diameter,  transferred  in  the  usual  way,  gives  the  major 
axis  of  the  ellipse  which  constitutes  the  outline  of  the  shadow  sought, 
while  its  minor  axis  is  at  once  determined  by  a  5,  equal  and  parallel  to 
AB. 

Fig.  10  exhibits  the  case  of  a  circle  parallel  to  the  vertical  plane  of  pro- 
jection, throwing  its  shadow  at  once  upon  two  plane  surfaces  inclined  to 
each  other.  To  delineate  this  shadow,  all  that  it  is  necessary  specially 
to  point  out  is,  that  the  points  d  and  e  are  found  by  drawing  from  Y  a 


318  SHADING  AND   SHADOWS. 

line  Y  D',  parallel  to  the  rays  of  light,  and  projecting  the  point  D'  to  D 
and  E. 

Fig.  12  represents  constructions  similar  to  the  foregoing,  for  obtaining 
the  form  of  the  shadow  cast  by  a  horizontal  circle  upon  a  vertical  curved 
surface. 

"We  may  here  remark,  that  in  every  drawing  where  the  shadows  are  to 
be  inserted,  it  is  of  the  utmost  importance  that  the  projections  which  re- 
present the  object  whose  shadow  is  required  should  be  exactly  defined,  as 
well  as  the  surface  upon  which  this  shadow  is  cast ;  it  is  therefore  advis- 
able, in  order  to  prevent  mistakes  and  to  insure  accuracy,  to  draw  the 
figures  in  China  ink,  and  to  erase  all  pencil  marks  before  proceeding  to 
the  operations  necessary  for  finding  the  shadows. 

Plate  LXXX.,  fig.  1. — To  find  the  outline  of  the  shadow  cast  upon  loth 
planes  of  projection  by  a  regular  hexagonal  pyramid. 

In  these  figures  it  is  at  once  obvious,  that  the  three  sides  A'  B'  F', 
A'  B'  C',  and  A'  C'  D'  alone  receive  the  light ;  consequently  the  edges 
A'  F'  and  A'  Df  are  the  lines  of  shade.  To  solve  this  problem,  then,  we 
have  only  to  determine  the  shadow  cast  by  these  two  lines,  which  is 
accomplished  by  drawing  from  the  projections  of  the  vertex  of  the  pyra- 
mid the  lines  A  b'  and  A'  a!  parallel  to  the  ray  of  light,  then  raising  from 
the  point  V  a  perpendicular  to  the  ground  line,  which  gives  at  a'  the 
shadow  of  the  vertex  on  the  horizontal  plane,  and  finally  by  joining  this 
last,  point  a'  with  the  points  D'  and  F' ;  the  lines  D'  a'  and  F'  a'  are  the 
outlines  of  the  required  shadow  on  the  horizontal  plane.  But  as  the  pyra- 
mid happens  to  be  situated  sufficiently  near  the  vertical  plane  to  throw  a 
portion  of  its  shadow  towards  the  vertex  upon  it,  this  portion  may  be 
found  by  raising  from  the  point  c,  where  the  line  A'  a'  cuts  the  ground-line, 
a  perpendicular  c  a,  intersecting  the  line  A  b'  in  aj  the  lines  a  d  and  a  e 
joining  this  point  with  those  where  the  horizontal  part  of  the  shadow 
meets  the  ground-line,  will  be  its  outline  upon  the  vertical  plane. 

Fig.  2. — Required  to  determine  the  limit  of  shade  on  a  cylinder  placed 
vertically ',  and  likewise  its  shadow  cast  upon  the  two  planes  of  projection. 

The  lines  of  shade  on  a  cylinder  situated  as  indicated,  are  at  once  found 
by  drawing  two  tangents  to  its  base,  parallel  to  the  ray  of  light,  and  pro- 
jecting through  the  points  of  contact  lines  parallel  to  the  axis  of  the 
cylinder. 

Draw  the  tangents  D'  d'  and  C'  c'  parallel  to  the  ray  E' ;  these  are  the 
outlines  of  the  shadow  cast  upon  the  horizontal  plane.  Through  the  point 
of  contact  C'  draw  the  vertical  line  C^C' ;  this  line  denotes  the  line  of  shade 
upon  the  surface  of  the  cylinder.  It  is  obviously  unnecessary  to  draw  the 


SHADING   AND    SHADOWS.  319 

perpendicular  from  the  opposite  point  D',  because  it  is  altogether  con- 
cealed in  the  vertical  elevation  of  the  solid.  In  order  to  ascertain  the 
points  C'  and  T)'  with  accuracy,  draw  through  the  centre  (y  a  diameter 
perpendicular  to  the  ray  of  light  R/. 

Had  this  cylinder  been  placed  at  a  somewhat  greater  distance  from  the 
vertical  plane  of  projection,  its  shadow  would  have  been  entirely  cast  upon 
the  horizontal  plane,  in  which  case  it  would  have  terminated  in  a  semi- 
circle drawn  from  the  centre  o' ',  with  a  radius  equal  to  that  of  the  base. 
But  as  a  portion  of  the  shadow  of  the  upper  part  is  thrown  upon  the  ver- 
tical plane,  its  outline  will  be  denned  by  an  ellipse  drawn  in  the  manner 
indicated  in  fig.  11,  plate  LXXIX.  • 

Fig.  3. — To  find  the  line  of  shade  in  a  reversed  cone,  and  its  shadow 
cast  upon  the  two  planes  of  projection. 

From  the  centre  A!  of  the  base  draw  a  line  parallel  to,  the  ray  of  light ; 
from  the  point  a',  where  it  intersects  the  perpendicular,  describe  a  circle 
equal  to  the  base,  and  from  the  point  A'  draw  the  lines  A'  V  and  A'  c',  touch- 
ing this  circle  ;  these  are  the  outlines  of  tlie  shadow  cast  upon  the  horizon- 
tal plane.  Then  from  the  centre  A'  draw  the  radii  A7  B'  and  A'  C'  parallel 
to  a!  V  and  a!  c'  /  these  radii  are  the  horizontal  projections  of  the  lines  of 
shade,  the  former  of  which,  transferred  to  B  D,  is  alone  visible  in  the  ele- 
vation. But  in  order  to  trace  the  outline  of  that  portion  of  the  shadow 
which  is  thrown  upon  the  vertical  plane,  it  is  necessary  to  project  the 
point  C'  to  C,  from  which,  by  a  construction  which  will  be  manifest  from 
inspection  of  the  figures,  we  derive  the  point  c  and  the  line  c  d  as  part  of 
the  cast  shadow  of  the  line  C'  A'.  The  rest  of  the  outline  of  the  vertical 
portion  of  the  cast  shadow  is  derived  from  the  circumference  of  the  base, 
as  in  fig.  2. 

Fig.  4. — "When  the  cylinder  is  placed  horizontally,  and  at  the  same 
time  at  an  angle  with  the  vertical  plane,  the  construction  is  the  same  as 
that  explained  (fig.  2) ;  namely,  lines  are  to  be  drawn  parallel  to  the  ray 
of  light,  and  touching  the  opposite  points  of  either  base  of  the  cylinder, 
and  through  the  points  of  contact  A  and  C  the  horizontal  lines  A  E  and 
C  D  are  to  be  drawn,  denoting  the  limits  of  the  shade  on  the  figure.  The 
latter  of  these  lines  only  is'  visible  in  the  elevation,  while,  on  the  other 
hand,  the  former,  A  E  alone,  is  seen  in  the  plan,  where  it  may  be  found 
by  drawing  a  perpendicular  from  A  meeting  the  base  F'  G'  in  A'.  The 
line  A7  E'  drawn  parallel  to  the  axis  of  the  cylinder  is  the  line  of  shade 
required.  Project  the  shadow  of  the  line  A  E  on  the  vertical  plane  as  in 
previous  examples,  and  the  construction  ^  will  define  the  outline  of  the 
shadow  of  the  cylinder. 


320  SHADING  AND  SHADOWS. 

The  example  here  given  presents  the  particular  case  in  which  the  base 
of  the  cylinder  is  parallel  to  the  direction  of  the  rays  of  light  in  the  hori- 
zontal projection.  In  this  case,  all  that  is  required  in  order  to  determine 
the  line  A'  E'  is  to  ascertain  the  angle  which  the  ray  of  light  makes  with 
the  projection  of  the  figure.  Draw  a  tangent  to  the  circle  F'  A2  G'  (which 
represents  the  base  of  the  cylinder  laid  down  on  the  horizontal  plane),  in 
such  a  manner  as  to  make  with  F'  G'  an  angle  of  35°  16',  and  through  the 
point  of  contact  A2  draw  a  line  parallel  to  the  axis  of  the  cylinder ;  this 
line  E'  A'  will  be  the  line  of  shade  as  before. 

Fig.  5  represents  a  cylinder  upon  which  a  shadow  is  thrown  by  a  rec- 
tangular prism,  of  which  the  sides  are  parallel  to  the  planes  of  projection. 
The  shadow  in  this  case  is  derived  from  the  edges  A'  D'  and  A'  E',  the 
first  of  which,  being  perpendicular  to  the  plane  of  projection,  gives,  accord- 
ing to  principles  already  laid  down,  a  straight  line  at  an  angle  of  45°  for 
the  outline  of  its  shadow,  whereas  the  side  A'  E'  being  parallel  to  that 
plane,  its  shadow  is  determined  by  a  portion  of  a  circle  a  b  <?,  described 
from  the  centre  o. 

Figs.  6,  7. — If  the  prism  be  hexagonal,  or  a  cylinder  be  substituted  for 
it,  the  mode  of  construction  remains  the  same.  But  it  should  be  observed, 
that  it  is  best  in  all  such  cases  to  commence  by  finding  the  points  which 
indicate  the  main  direction  of  the  outline.  To  ascertain  the  point  a  at 
which  the  shadow  commences,  draw  from  a'  the  line  a'  A'  at  an  angle  of 
45°,  which  is  then  to  be  projected  vertically  to  a  A.  Then  the  highest 
point  5  (fig.  7)  should  be  determined  by  the  intersection  of  the  radius 
O'  B'  (drawn  parallel  to  the  ray),  with  the  circumference  of  the  base  of 
the  cylinder  on  which  the  required  shadow  is  cast ;  and  finally,  the  point 
c,  where  the  outline  of  the  cast  shadow  intersects  the  line  of  shade,  should 
be  determined  by  a  similar  process. 

Fig.  1  represents  a  hexagonal  prism  upon  which  a  shadow  is  thrown 
by  a  rectangular  prism.  Determine  the  point  a  as  in  fig.  5,  pi.  LXXX. ; 
draw  from  the  angular  points  J',  c'  lines  parallel  to  the  direction  of  the 
light,  intersecting  the  edge  of  the  rectangular  prism  at  B',  C' ;  project  these 
points,  and  draw  the  lines  B  5,  C  c/  their  intersections  with  the  edges  of 
the  hexagonal  prism  will  be  the  limiting  points  &,  <?,  of  the  required 
shadow. 

Fig.  2  represents  a  hexagonal  prism  upon  which  a  shadow  is  cast  by 
another  hexagonal  prism.  The  construction  is  precisely  similar  to  the 
preceding.  Lines  parallel  to  the  direction  of  the  light  are  drawn  from 
the  angular  points  of  both  interior  and  exterior  prisms ;  these  points  are 
projected,  and  the  limiting  points  «,  1>,  c,  d,  of  the  shadow  are  determined. 


SHADING   AND    SHADOWS. 


321 


Fig.  3  represents  a  hexagonal  prism  upon  which  a  shadow  is  cast  by  a 
cylinder,  a  variety  of  the  preceding;  but  as  in  this  case  the  outline 
of  the  shadow  is  curved,  in  addition  to  the  lines  from  the  angular  points 


Fig.  1. 


Fig.  2. 


Fig.  8. 


of  the  prism,  parallels  are  also  drawn  from  as  many  intermediate  points 
b'  df,  as  may  be  necessary  to  determine  the  outline  of  the  curved  shadow. 

Plate  LXXXL,  fig.  7. — To  define  the  shadows  cast  upon  tJie  interior  of 
a  hollow  cylinder  in  section  by  itself  and  by  a  circular  piston  fitted  into  it. 

The  example  shows  a  steam  cylinder,  A,  in  section,  by  a  plane  passing 
through  its  axis,  with  its  piston  and  rod  in  full. 

Conceive,  in  the  first  instance,  the  piston  P  to  be  removed  ;  the  shadow 
cast  into  the  interior  of  the  cylinder  will  then  consist,  obviously,  of  that 
projected  by  the  vertical  edge  B  C,  and  by  a  portion  of  the  horizontal 
edge  B  A.  To  find  the  first,  draw  through  B7  a  line  B'  V  at  an  angle  of 
45°  with  B7  A ;  the  point  57,  where  this  line  meets  the  interior  surface  of 
the  cylinder,  being  projected  upwards  to  fig.  7,  gives  the  line  bf  as  the 
outline  of  the  shadow  sought.  Then,  parallel  to  the  direction  of  the  light, 
draw  a  tangent  at  F7  to  the  inner  circle  of  the  base  ;  its  point  of  contact 
being  projected  to  F  in  the  elevation,  marks  the  commencement  of  the  out- 
line of  the  shadow  cast  by  the  upper  edge  of  the  cylinder.  The  point  5, 
where  it  terminates,  will  obviously  be  the  intersection  of  the  straight  line 
/  b  already  determined,  with  a  ray  B  b  from  the  tipper  extremity  of  the 
edge  B  C  ;  and  any  intermediate  point  in  the  curve,  as  e,  may  be  found  by 
taking  a  point  E7,  between  B7  and  57,  projecting  it  to  E,  and  causing  rays 
E  <?,  E'  e',  to  pass  through  these  points.  The  outline  of  the  shadow  required 
will  then  be  the  curve  F  e  b  and  the  straight  line  bf.  'Suppose  now  the 
piston  P  and  its  rod  T  to  be  inserted  into  the  cylinder  as  shown.  The 
lower  surface  of  the  piston  will  then  cast  a  shadow  upon  the  interior  sur- 
face of  the  cylinder,  of  which  the  outline  D  d  h  o,  may  be  formed  in  the 
21 


322 


SHADING   AND   SHADOWS. 


same  way,  as  will  be  obvious  from  inspection  of  the  figures,  and  compari- 
son of  the  letters  of  reference.  The  piston-rod  T  being  cylindrical  and 
vertical,  it  casts  also  its  shadow  into  the  interior  of  the  cylinder ;  it  will 
obviously  consist  of  a  rectangle  ij  I  lc  drawn  parallel  to  the  axis,  and  of 
which  the  sides  ij  and  &  I  are  determined  by  the  tangents  I'  i'  and  K/  Icf. 

Figure  4. — This  example  consists  of  a  hollow  cylinder,  surmounted 
by  a  circular  disc  or  cover,  sectioned  through  the  centre,  where  it  is  also 
penetrated  by  a  cylindrical  aperture.  The  construction  necessary  for  find- 
ing the  outlines  of  the  cast  shadow  will  obviously  be  the  same  as  already 
laid  down.  In  this  case,  however,  it  is  proper  to  know  beforehand  what 
parts  of  the  upper  and  lower  edges  of  the  central  aperture  cast  their 
shadows  into  the  interior  of  the  cylinder ;  if,  then,  we  take  the  trouble  to 
construct  the  shadows  of  each  of  these  edges  separately,  we  shall  find  that 
that  of  the  upper  edge  is  a  curve  1}  cf,  and  that  of  the  lower  a  similar 
curve  a  c  e,  cutting  the  former  in  c.  This  point  limits  the  parts  of  each 
curve  which  are  actually  visible  ;  namely,  the  portion  5  c  of  the  first,  and 
the  portion  e  c  of  the  second  ;  hence  it  follows,  that  in  order  to  avoid  un- 
necessary work,  we  should  first  determine  the  position  of  the  point  of  inter- 
section, c,  of  the  two  curves,  which  is  in  fact  the  cast  shadow  of  the  lowest 
point  0  in  the  curve  D  C,  previously  laid  down  in  the  circular  opening  of 
the  cover,  in  the  manner  indicated  in  fig.  7,  plate  LXXXI. 


Fig.  4. 


Fig.  r,. 


Fig.  5.-Let  a  cylinder  in  section  to  be  set  at  an  inclination  to  the  horizon- 
tal plane.  To  find  the  outline  of  the  shadow  cast  into  its  interior,  describe 
upon  the  prolongation  of  the  axis  of  the  cylinder  a  semicircle  A'  a  B', 
representing  its  interior  surface,  and  then,  in  any  convenient  part  of  the 
paper,  draw  the  diagonal  in  o  parallel  to  the  line  of  light  A'  Er,  and  con- 


SHADING  AND  SHADOWS. 


323 


struct  a  square  in  n  op  (fig.  6) ;  from  one  of  the  extremities,  o,  draw  the 
line  o  r  parallel  to  A'  B',  and  through  the  opposite  extremity,  m,  draw  a 
perpendicular  r  s  to  this  line,  and  set  off  on  the  perpendicular  the  distance 
r  s  equal  to  the  side  of  the  square,  and  join  s  o.  Now, 
draw  through  the  point  A',  in  the  original  figure,  a  line 
A'  «',  parallel  to  s  o,  intersecting  the  circle  A'  a!  B'  in  the 
point  «',  which  being  projected  by  a  line  parallel  to  the 
axis  of  the  cylinder,  and  meeting  the  line  A  «,  drawn  at 
an  angle  of  45°,  gives  the  first  point  a  in  the  curve  C  d  a. 
The  other  .points  will  be  obtained  in  like  manner,  by  draw- 
ing at  pleasure  other  lines,  such  as  T)'  d',  parallel  to  A'  a'.  Flg.  6. 

To  find  the  outline  of  the  shadow  cast  into  the  interior  of  a  hollow 
hemisphere. 

Let  A  B  C  D  (fig.  7)  represent  the  horizontal  projection  of  a  concave 
hemisphere.  Here  it  is  sufficiently 
obvious,  that  if  we  draw  through 
the  centre  of  the  sphere  a  line  per- 
pendicular to  the  ray  of  light  A  C, 
the  points  B  and  D  will  at  once 
give  the  extremities  of  the  curve 
sought.  Construct  the  square  (fig. 
6),  making  m  o  the  diagonal  paral- 
lel witli  A  C'  at  m  ;  erect  the  per- 
pendicular in  m3,  making  m  m3 
equal  to  one  side  of  the  square ; 
connect  m3  o.  Take  now  upon  the 
prolongation  of  the  line  B  D  any 
point  O',  from  which,  as  a  centre,  describe  a  semicircle  with  the  radius 
A  O,  and  from  the  point  A'  draw  the  straight  line  A'  a  parallel  to  om3; 
the  point  a!  of  its  intersection  with  the  circle  A'  a!  C',  projected  to  «,  will 
be  another  principal  point  in  the  outline  of  the  shadow. 

By  imagining  similar  sections,  such  as  E  F  parallel  to  the  former  A  C, 
and  laying  down  in  the  same  way  semicircles  E  F  corresponding  to  them, 
and  drawing  through  E7  lines  parallel  to  o  in3,  and  projecting  their  intersec- 
tion with  their  semicircles  to  the  corresponding  sections  E  F  on  the  plan,  the 
remaining  points  in  the  curve  sought  may  be  obtained.  But  as  this  curve 
is  an  ellipse,  of -which  the  diameter  D  B  is  the  major  axis,  and  the  line  0  a 
the  half  minor  axis,  it  follows  that  this  last  line  being  determined,  the 
curve  may  be  constructed  by  the  ordinary  methods  for  ellipses. 


Fig.  T. 


324: 


SHADING   AND   SHADOWS. 


There  will  now  be  no  difficulty  in  constructing  the  cast  shadow  in  the 
interior  of  a  concave  surface  (fig.  8),  formed  by  the  combination  of  a  hollow 
semi-cylinder  and  a  quadrant  of  a  hollow  sphere,  called  a  niche,  as  we 
know  the  mode  of  tracing  the  shadows  upon  each  of  these  figures  separ- 
ately. Thus,  the  shadow  of  the  circular  outline  upon  the  spherical  por- 
tion is  part  of  an  ellipse 
i  c  D,  whose  semi- axis  major 
is  O  D ;  the  semi-axis  minor 
is  obtained  by  describing  the 
semicircle  B2  i'  E  with  the 

>  y'\    ,'•  • .—••*'       V 

;xv.-V  r   radius  O  B,  drawing   from 

the  point  B2  the  straight  line 
B2  i'  parallel  to  a  line  o  m3, 
found  by  a  construction  as  before  (fig.  6), 
and  finally  projecting  the  point  of  intersec- 
tion i'  to  i  on  the  straight  line  B  O.  The 
point  e,  where  this  ellipse  cuts  the  horizontal 
diameter  A  F,  limits  the  cast  shadow  upon 
the  spherical  surface  ;  therefore  all  the  points 
beneath  it  must  be  determined  upon  the  cy- 
lindrical part,  Through  A'  in  the  plan  draw 
the  line  A'  a'  parallel  to  the  ray  of  light ; 
project  a'  till  it  intersects  the  line  of  light 
A  a  in  the  elevation  at  a.  The  line  of  shadow 
below  a  is  the  shadow  of  the  edge  of  the  cy- 
linder, and  must  therefore  be  a  straight  line. 
The  line  of  shadow  between  a  and  e  is  pro- 
duced by  the  outline  of  the  circular  part  fall- 
ing on  a  cylindrical  surface,  and  is  established  as  in  previous  constructions, 
by  drawing  lines  parallel  to  the  rays  of  light  through  different  points,  as 
B  in  the  curved  outline,  and  similar  lines  through  the  corresponding 
points  B7  in  the  plan,  and  projecting  their  intersections  V  with  the  semir 
circle  till  they  intersect  the  first  line  at  &  as  points  in  the  line  of  shadow." 

Plate  LXXXL,  figs.  1,  2. — To  find  the  line  of  shade  in  a  sphere.,  and 
the  outline  of  its  shadow  cast  upon  the  horizontal  plane. 

The  line  of  shade  in  a  sphere  is  simply  the  circumference  of  a  great 
circle  of  which  the  plane  is  perpendicular  to  the  direction  of  the  luminous 
ray,  and  consequently  inclined  to  the  two  planes  of  projection.  This  line 
will,  therefore,  be  represented  in  elevation  and  plan  by  two  equal  ellipses, 


Fig.  8. 


SHADING  AND   SHADOWS.  325 

the  major  axes  of  which  are  obviously  the  diameters  C.D  and  C'  D',  drawn 
at  an  angle  of  45°. 

To  find  the  minor  axes  of  these  curves,  assume  any  point  O2,  upon  the 
prolongation  of  the  diameter  of  the  perpendicular  C'  D'  (fig.  2),  draw 
through  this  point  the  straight  line  O2  o',  inclined  at  an  angle  of  35°  16', 
to  A'  B'  or  its  parallel,  and  erect  upon  it  the  perpendicular  E2  F2.  The 
projection  of  the  two  extremities  E2  and  F2  upon  the  line  A'  B',  will  give, 
in  the  plan,  the  line  E7  F7  for  the  length  of  the  required  minor  axis  of  the 
ellipse,  i.  e.  of  the  line  of  shade  in  the  plan ;  and  this  line  being  again 
transferred  to  the  elevation,  determines  the  minor  axis  E  F  of  the  line  of 
shade  in  the  elevation. 

Supposing  it  were  required  to  draw  these  ellipses,  not  by  means  of 
their  axes,  but  by  points,  any  number  of  these  may  be  obtained  by  making 
horizontal  sections  of  the  sphere.  Thus,  for  example,  if  we  draw  the  chord 
G  H,  parallel  to  A'  B',  to  represent  one  of  these  sections,  and  from  the 
point  a,  where  it  cuts  the  diameter  E2  F2,  if  we  draw  a  perpendicular  to 
A7  B',  the  points  a'  a',  where  it  intersects  the  circumference  of  a  circle 
representing  the  section  G  H  in  plan,  will  be  two  points  in  the  line  of 
shade  required.  These  points  may  be  transferred  to  the  elevation,  by 
supposing  a  section  g  h  to  be  made  in  fig.  1,  corresponding  to  G  H  in  fig. 
2,  and  projecting  the  points  a!  a!  by  perpendiculars  to  g  h,  the  line  repre- 
senting the  cutting  plane. 

The  outline  of  the  shadow  cast  by  the  sphere  upon  the  horizontal  plane 
is  also  obviously  an  ellipse  ;  it  may  be  constructed  either  by  means  of  its 
two  axes,  or  by  the  help  of  points,  in  the  manner  indicated  in  the  figure. 

Figs.  3,  4,  and  5. — To  draw  the  line  of  shade  on  the  surface  of  a  ring 
of  circular  section,  in  vertical  section,  elevation,  and  plan. 

We  shall  first  point  out  the  mode  of  obtaining  those  primary  points  in 
the  curve  which  are  most  easily  found,  and  then  proceed  to  the  general 
case  of  determining  any  point  whatever.  • 

If  tangents  be  draAvn  to  the  circles  represented  in  figs.  3  and  4,  parallel 
to  the  ray  of  light,  their  points  of  contact,  a,  I,  c,  d,  will  be  the  starting 
points  of  the  required  lines  of  shade. 

Again,  the  intersections  of  'the  horizontal  lines  a  e,  d  g,  c  /,  drawn 
through  these  points,  with  the  axis  of  the  ring,  will  give  so  many  new 
points  <?,  g,f,  in  the  curve.  These  points  are  denoted  in  the  plan  (fig.  5),  by 
setting  off  the  distances  a  e  and  c  f  upon  the  vertical  line  g'  D,  on  both 
sides  of  the  centre  C'. 

Farther,  the  diameter  F7  G',  drawn  at  an  angle  of  45°,  determines,  by 
its  intersections  with  the  exterior  and  interior  circumferences  of  the  ring, 


326  SHADING  AND  SHADOWS. 

four  other  points  F',  t' ,  a?',  and  G',  in  the  curve  in  question  ;  these  points 
are  all  to  be  projected  vertically  upon  the  line  A  B. 

And,  lastly,  to  obtain  the  lowest  points  I  /,  construct  the  two  squares 
(fig.  6),  making  the  diagonals  a'  mf  and  o  rn  parallel  severally  to  the  ray 
of  light  in  plan  and  elevation ;  revolve  o'  m  upon  the  point  of  until  it  be- 
comes parallel  with  the  vertical  plane  of  projection;  project  to  m2,  and 
connect  o  m? ;  draw  tangents  to  the  circles  represented  in  figs.  3  and  4, 
parallel  to  the  line  o  m2,  and  transfer  the  distances  between  the  points  of 
contact,  s,  s,  and  the  axis  of  the  ring,  to  the  radius  E'  C'  (fig.  5),  where 
they  are  denoted  by  I'  I'  /  these  latter  points  are  then  to  be  projected  ver- 
tically to  Z,  I,  upon  the  horizontal  lines  drawn  through  the  same  points  s,  s 
(figs.  3  and  4). 

The  curves  sought  might  now,  in  most  instances,  be  traced  by  the 
points  thus  obtained ;  but  should  the  ring  be  on  a  large  scale,  and  great 
accuracy  be  required,  it  may  be  proper  to  determine  a  greater  number 
of  points.  For  this  purpose,  draw  through  the  centre  C',  a  straight  line 
I'  H',  in  any  direction,  draw  through  o',  one  of  the  angular  points  of  the 
cube  at  fig.  6,  a  straight  line  or  parallel  to  I'  IF,  and  from  the  opposite 
point  tn'  draw  a  perpendicular  m'  r'  to  o'  T'.  Then,  having  revolved  the 
point  r'-  to  r1  by  means  of  a  circular  arc,  in  order  to  admit  of  this  last 
point  being  projected  to  ?',  we  join  o  T. 

Applying  this  construction  to  the  figures  before  us,  we  now  draw  tan- 
gents to  the  circles  represented  in  figs.  3  and  4,  parallel  to  the  line  o  r, 
and,  taking  as  radii  the  distances  from  their  respective  points  of  contact, 
h  and  I,  to  the  axis  of  the  ring,  we  describe  corresponding  circles  about 
the  centre  C',  fig.  4.  We  thus  obtain  four  other  points  in  the  curves  re- 
quired, namely,  I',  *',  A,  and  H,  which  may  also  be  projected  upon  the 
horizontal  lines  drawn  through  the  points  h  or  I. 

By  drawing  the  straight  line  J'  K'  so  as  to  form  with  F'  G'  the  same 
angle  which  the  latter  makes  with  the  line  H'  I',  we  obtain,  by  the  inter- 
section of  that  line  with  the  circles  last  named,  ftfur  other  points  of  the 
curves  in  question. 

Figs.  8  and  9. — Of  the  shadows  cast  upon  the  surfaces  of  grooved  pulleys. 

The  construction  of  cast  shadows  upon  surfaces  of  the  kind  now  under 
consideration  is  founded  upon  the  principle  already  announced,  that  when 
a  circle  is  parallel  to  a  plane,  its  shadow ,  cast  upon  that  plane,  is  another 
circle  equal  to  the  original  circle. 

Take,  in  the  first  place,  the  case  of  a  circular-grooved  pulley  (figs.  8 
and  9) ;  the  cast  shadow  on  its  surface  is  obviously  derived  from  the  cir- 
cumference of  the  upper  edge  A  B.  To  determine  its  outline,  take  any 


SHADING  AND  SHADOWS.  327 

horizontal  line  D  E  upon  fig.  8,  and  describe  from  the  centre  C'  (fig.  9)  a 
circle  with  a  radius  equal  to  the  half  of  that  line  ;  then  draw,  through  the 
same  centre,  a  line  parallel  to  the  ray  of  light,  which  will  intersect  the 
plane  D  E  in  c  ;  lastly,  describe  from  the  point  c',  as  a  centre,  an  arc  of  a 
circle  with  a  radiifs  equal  to  A  C  ;  the  point  of  intersection,  #',  of  this  arc, 
with  the  circumference  of  the  plane  D  E,  will  give  when  projected  to  a 
(fig.  8),  one  of  the  points  in  the  curve  required. 

To  avoid  unnecessary  labor  in  drawing  more  lines  parallel  to  D  E  than 
are  required,  it  is  important,  in  the  first  place,  to  ascertain  the  highest 
point  in  the  curve  sought  This  point  is  the  shadow  of  that  marked  H  on 
the  upper  edge  of  the  pulley,  and  which  is  determined  by  the  intersection 
of  the  ray  C'  H'  with  the  circumference  of  that  edge  in  the  plan  ;  and  it 
is  obtained  by  drawing  through  the  point  A  (fig.  8)  a  straight  line  at  an 
angle  of  35°  16'  with  the  line  A  B,  and  through  the  point  e,  striking  a 
horizontal  line  e.f,  which  by  its  intersection  with  the  line  H  A,  drawn  at 
an  angle  of  45°,  will  give  the  point  sought. 

In  fig.  9,  the  pulley  is  supposed  to  be  divided  horizontally  in  the  centre, 
and  the  shadow  represented  is  derived  from  the  smaller  circle  I  K,  and  is 
easily  constructed  by  methods  above  described. 

Plate  LXXXII. — To  trace  the  outlines  of  the  shadows  cast  upon  the 
surfaces  of  screws  and  nuts,  both  triangular  and  square-threaded. 

Figs.  1  and  2  represent  the  projections  of  a  screw  with  a  single  square 
thread,  and  placed  in  a  horizontal  position,  A'  a'  being  the  direction  of  the 
ray  of  light.  In  this  example,  the  shadow  to  be  determined  is  simply  that 
cast  by  the  outer  edge,  A  B,  of  the  thread  upon  the  surface  of  the  inner 
cylinder ;  therefore  its  outline  is  to  be  delineated  in  the  same  manner  as  we 
have  already  pointed  out,  in  treating  of  a  cylinder  surmounting  another 
of  smaller  diameter  (page  320). 

Figs.  3  and  4. — The  case  of  a  triangular-threaded  screw  does  not  admit 
of  so  easy  a  solution  as  the  above,  because  the  outer  edge  A  C  D  of  the 
thread,  in  place  of  throwing  its  shadow  upon  a  cylinder,  projects  it  upon  a 
helical  surface  inclining  to  the  left,  of  which  the  .generatrix  is  known.  De- 
scribe from  the  centre  O  (fig.  3)  a  number  of  circles,  representing  the 
bases  of  so  many  cylinders,  on  the  surfaces  of  which  we  must  suppose 
helical  lines  to  be  traced,  of  the  same  pitch  with  those  which  form  the  ex- 
terior edges  of  the  screw  (see  fig.  4).  We  must  now  draw  any  line,  such 
as  B'  E',  parallel  to  the  ray  of  light,  and  cutting  all  the  circles  described 
in  fig.  3  in  the  points  B',  F',  G',  E7,  which  are  then  to  be  successively  pro- 
jected to  their  corresponding  helical  lines  in  fig.  4,  where  they  are  denoted 
by  B2,  F,  G,  and  E.  Then,  transferring  the  point  B'  (fig.  3)  to  its  appro- 


328  SHADING  AND  SHADOWS. 

priate  position  B  on  the  edge  A  C  D  (fig.  4),  and  drawing  through  the 
latter  a  line  B  5  at  an  angle  of  45°,  its  intersection  with  the  curve  B2  G  E 
will  give  one  point  in  the  curve  of  the  shadow  required.  In  the  same 
manner,  by  constructing  other  curves,  such  as  H2  J  K,  the  remaining 
points,  as  A,  in  the  curve  may  be  found. 

Figs.  5  and  6. — The  same  processes  are  requisite  in  order  to  determine 
the  outlines  of  the  shadows  cast  into  the  interior  surfaces  of  the  nut  cor- 
^  responding  to  the  screw  last  described,  as  will  Be  evident  from  inspection 
of  figs.  5  and  6.  These  shadows  are  derived  not  only  from  the  helical 
edge  A  B  D,  but  also  from  that  of  the  generatrix  A  C. 

Figs.  7  and  8. — The  shadow  cast  by  the  helix  ABC  upon  the  concave 
surface  of  the  square-threaded  nut  is  a  curve  a  ~b  C,  which  is  to  be  deter- 
mined in  the  same  way  as  that  in  the  interior  of  a  hollow  cylinder.  The 
same  observation  applies  to  the  edges  A  A2  and  A2  E,  as  well  as  to  those 
of  the  helix  F  G  H  and  the  edge  H  I.  With  regard  to  the  shadow  of  the 
two  edges  J  K  and  Iv  L,  they  must  obviously  follow  the  rules  laid  down 
in  reference  to  figs.  4  and  6,  seeing  that  it  is  thrown  upon  an  inclined  heli- 
cal surface,  of  which  A  L  is  the  generatrix. 

The  principles  so  fully  laid  down  and  illustrated  in  the  preceding  pages 
will  be  found  to  admit  of  a  ready  and  simple  application  to  the  delineation 
of  the  shadows  of  all  the  ordinary  forms  and  combinations  of  machinery 
and  architecture,  however  varied  or  complicated  ;  and  the  student  should 
exercise  himself,  at  this  stage  of  his  progress,  in  tracing,  according  to  the 
methods  above  explained,  the  outlines  of  the  cast  shadows  of,pulleys,  spur- 
wheels,  and  such  simple  and  elementary  pieces  of  machinery.  It  must  be 
observed,  that  the  student  should  never  copy  the  figures  as  here  repre- 
sented, but  should  adopt  some  convenient  scale  somewhat  larger  than  our 
figures,  and  construct  his  drawings  according  to  the  description,  looking  to 
the  figures  as  mere  illustrations ;  in  this  way  the  principles  of  the  con- 
struction will  be  more  surely  understood,  and  more  firmly  fixed  in  his 
mind. 

MANIPULATION   OF   SHADING  AND   SHADOWS.— METHODS   OF   TINTING. 

The  intensity  of  .a  shade  or  shadow  is  regulated  by  the  various  peculi- 
arities in  the  forms  of  bodies,  and  by  the  position  which  objects  may 
occupy  in  reference  to  the  light. 

Surfaces  in  the  light. — Flat  surfaces  wholly  exposed  to  the  light,  and 
at  all  points  equidistant  from  the  eye,  should  receive  a  uniform  tint. 

In  geometrical  drawings,  where  the  visual  rays  are  imagined  parallel 
to  the  plane  of  projection,  every  surface  parallel  to  this  plane  is  supposed 


SHADING  AND  SHADOWS.  329 

to  have  all  its  parts  at  the  same  distance  from  the  eye ;  such  is  the  vertical 
side  of  the  prism  a  led  (fig.  4,  plate  LXXXIII). 

When  two  surfaces  thus  situated  are  parallel,  the  one  nearer  the  eye 
should  receive  a  lighter  tint  than  the  other.  Every  surface  exposed  to  the 
light,  but  not  parallel  to  the  plane  of  projection,  and  therefore  having  no 
two  points  equally  distant  from  the  eye,  should  receive  an  unequal  tint. 
In  conformity,  then,  with  the  preceding  rule,  the  tint  should  gradually  in- 
crease in  depth  as  the  parts  of  such  a  surface  recede  from  the  eye.  This 
"  effect  is  represented  in  the  same  figure  on  the  surface,  a  dfe,  which,  by 
reference -to  the  plan  (fig.  1),  is  found  to  be  in  an  inclined  position. 

If  two  surfaces  are  unequally  exposed  to  the  light,  the  one  which  is 
more  directly  opposed  to  its  rays  should  receive  the  fainter  tint. 

Tims  the  face  e'  a'  (fig.  1),  presenting  itself  more  directly  to  the  rays  of 
light  than  the  face  a'  5',  receives  a  tint  which,  although  graduated  in  con- 
sequence of  the  inclination  of  this  face  to  the  plane  of  projection,  becomes 
at  that  part  of  the  'surface  situated  nearest  to  the  eye  fainter  than  the  tint 
on  the  surface  a  5. 

Surfaces  in  shade. — When  a  surface  entirely  in  the  shade  is  parallel  to 
the  plane  of  projection,  it  should  receive  a  uniform  dark  tint. 

"When  two  objects  parallel  to  each  other  are  in  the  shade,  the  one 
nearer  the  eye  should  receive  the  darker  tint. 

"When  a  surface  in  the  shade  is  inclined  to  the  plane  of  projection,  those 
parts  which  are  nearest  to  the  eye  should  receive  the  deepest  tint. 

The  face  ~b  g  ti  c  (fig.  4),  projected  horizontally  at  V  g'  (fig.  1),  is  situated 
in  this  manner.  It  will  there  be  seen,  that  towards  the  line  I  c  the  tint  is 
much  darker  than  it  is  where  it  approaches  the  line  g  h. 

If  two  surfaces  exposed  to  the  light,  but  unequally  inclined  to  its  rays, 
have  a  shadow  cast  upon  them,  that  part  of  it  which  falls  upon  the  surface 
more  directly  influenced  by  the  light  should  be  darker  than  where  it  falls 
upon  the  other  surface. 

Exemplifications  of  the  foregoing  rules  may  be  seen  on  various  figures 
in  the  plates. 

In  order  that  these  rules  may  be  practised  with  proper  effect,  we  shall 
give  some  directions  for  using  the  brush  or  hair-pencil,  and  explain  the 
usual  methods  employed  for  tinting  and  shading. 

The  methods  of  shading  most  generally  adopted  are  either  by  the 
superposition  of  any  number  of  flat  tints,  or  by  tints  softened  off  at  their 
edges.  The  former  method  is  the  more  simple  of  the  two,  and  should  bo 
the  first  attempted. 

Shading  ~by  flat  tints. — Let  it  be  proposed  to  shade  the  prism  (fig.  4, 


330  SHADING  AXD  SHADOWS. 

plate  LXXXIII.),  by  means  of  flat  tints.  According  to  the  position  of  the 
prism,  as  shown  by  its  plan  (fig.  1),  the  face  a  1)  c  d  (fig.  4)  is  parallel  to  the 
plane  of  projection,  and  therefore  entirely  in  the  light.  This  face  should 
receive  a  uniform  tint  either  of  India  ink  or  sepia.  When  the  surface  to  be 
tinted  happens  to  be  very  large,  it  is  advisable  to  put  on  a  very  light  tint 
first,  and  then  to  go  over  the  surface  a  second  time  with  a  tint  sufficiently 
dark  to  give  the  desired  tone  to  the  surface. 

The  face  l>g he  being  inclined  to  the  plane  of  projection,  as  is  shown 
by  the  line  V  g'  in  the  plan  (fig.  1),  should  receive  a  graduated  tint  from ' 
the  line  1)  c  to  the  line  g  h.  This  graduality  is  obtained  by  laying  on  a 
succession  of  flat  tints  in  the-  following  manner : — First,  divide  the  line 
V  g'  (fig.  1)  into  equal  parts  at  the  points  1',  2',  and  from  these  points  pro- 
ject lines  upon,  and  parallel  to,  the  sides  of  the  face  b  g  h  c  (fig.  4).  These 
lines  should  be  drawn  very  lightly  in  pencil,  as  they  merely  serve  to  cir- 
cumscribe the  tints.  A  greyish  tint  is  then  spread  over  that  portion  of  the 
face  l>ghc  (fig.  2),  between  the  lines  l>  c  and  1,  1.  When  this  is  dry,  a 
similar  tint  is  to  be  laid  on,  extending  over  the  space  comprised  within  the 
lines  1)  c  and  2,  2  (fig.  3).  Lastly,  a  third  tint  covering  the  whole  surface 
1}  c  h  g  (fig.  4)  imparts  the  desired  graduated  shade  to  that  side  of  the  prism. 
The  number  of  tints  designed  to  express  such  a  graduated  shade  depends 
upon  the  size  of  the  surface  to  be  shaded ;  and  the  depth  of  tint  must  vary 
according  to  this  number. 

As  the  number  of  these  washes  is  increased,  the  whole  shade  gradually 
presents  a  softer  appearance,  and  the  lines  which  border  the  different  tints 
become  less  harsh  and  perceptible.  For  this  reason  the  foregoing  method 
of  representing  a  shade  or  graduated  tint  by  washes  successively  passing 
over  each  other  is  preferable  to  that  sometimes  employed,  of  first  covering 
the  whole  surface  lygJic  with  a  faint  tint,  then  putting  on  a  second  tint 
1)  2  2  c,  followed,  lastly,  by  a  narrow  wash  5  1  1  cy  because,  in  following 
this  process,  the  outline  of  each  wash  remains  untouched,  and  presents,  un- 
avoidably, a  prominence  and  harshness  which,  by  the  former  method,  are 
in  a  great  measure  subdued. 

The  face  adfe  is  also  inclined  to  the  plane  'of  projection,  as  shown  by 
the  line  a'  e'  in  the  plan  (fig.  1)  ;  but  as  it  is  entirely  in  the  light,  it  should 
be  covered  by  a  series  of  much  fainter  tints  than  the  surface  Iff  he,  which 
is  in  the  shade,  darkening,  however,  towards  the  line  ef.  The  gradation 
of  tint  is  effected  in  the  same  way  as  on  the  face  1)  g  he. 

Let  it  be  proposed  to  shade  a  cylinder  (fig.  12),  by  means  of  flat  tints  : 

In  shading  a  cylinder,  it  will  be  necessary  to  consider  the  difference  in 
the  tone  proper  to  be  maintained  between  the  part  in  the  light  and  that  in 


SHADING   AND   SHADOWS.  331 

the  shade.  It  should  be  remembered  that  the  line  of  separation  between 
the  light  and  shade  a  5  is  .determined  by  the  radius  O  a'  (fig.  5),  drawn 
perpendicular  to  the  rays  of  light  R  O.  That  part,  therefore,  of  the  cylin- 
der which  is  in  the  shade  is  comprised  between  the  lines  a  l>  and  c  d.  This 
portion,  then,  should  be  shaded  conformably  to  the  rule  previously  laid 
down  for  treating  surfaces  in  the  shade  inclined  to  the  plane  of  projection. 
All  the  remaining  part  of  the  cylinder  which  is  visible  presents  itself  to 
the  light ;  but,  in  consequence  of  its  circular  figure,  the  rays  of  light  form 
angles  varying  at  every  part  of  its  surface,  and  consequently  this  surface 
should  receive  a  graduated  tint.  In  order  to  represent  with  effect  the 
rotundity,  it  will  be  necessary  to  determine  with  precision  the  part  of  the 
surface  which  is  most  directly  affected  by  the  light.  This  part,  then,  is 
situated  about  the  line  e  i  (fig.  12),  in  the  vertical  plane  of  the  ray  of  light 
K  O  (fig.  5).  As  the  visual  rays,  however,  are  perpendicular  to  the  ver- 
tical plane,  and  therefore  parallel  to  V  O,  it  follows  that  the  part  which 
appears  clearest  to  the  eye  will  be  near  this  line  Y  O,  and  may  be  limited 
by  the  line  T  O,  which  bisects  the  angle  Y  O  E  and  the  line  K  O.  By 
projecting  the  points  e'  and  m',  and. drawing  the  lines  e  i  and  m  n  (fig.  12), 
the  surface  comprised  between  these  lines  will  represent  the  lightest  part 
of  the  cylinder. 

This  part  should  have  no  tint  upon  it  whatever  if  the  cylinder  happen 
to  be  polished  :  a  turned  iron  shaft  or  a  marble  column  for  instance  ;  but 
if  the  surface  of  the  cylinder  be  rough,  as  in  the  case  of  a  cast-iron  pipe, 
then  a  very  light  tint — considerably  lighter  than  on  any  other  part — may 
be  given  it. 

Again,  let  us  suppose  the  half-plan  of  the  cylinder/7  m'  a'  c'  (fig.  5),  to 
be  divided  into  any  number  of  equal  parts.  Indicate  these  divisions  upon 
the  surface  of  the  cylinder  by  faint  pencil  lines,  and  begin  the  shading  by 
laying  a  tint  over  all  that  part  of  the  cylinder  in  the  shade  a  c  d  l>  (fig.  6). 
This  will  at  once  render  evident  the  light  and  dark  parts  of  the  cylinder. 
When  this  is  dry,  put  on  a  second  tint  covering  the  line  a  5  of  separation 
of  light  and  shade,  and  extending  oArer  one  division,  as  shown  in  fig.  7. 
A  third  tint  should  be  spread  over  this  division,  and  one  on  each  side  of 
it,  as  in  fig.  8.  Proceed  in  this  way  until  the  whole  of  that  part  of  the 
cylinder  which  is  in  the  shade  is  covered.  The  successive  stages  of  this 
process  may  be  seen  in  figs.  9,  10,  and  11. 

Treat  in  a  similar  manner  the  part  feig,  and  complete  the  operation 
by  covering  the  whole  surface  of  the.  cylinder — excepting  only  the  division 
e  m  n  i  (fig.  12) — with  a  very  light  tint ;  the  cylinder  will  then  assume  the 
appearance  presented  by  fig.  12. 


3o2  SHADING      m>   SHADOWS. 

Shading  by  softened  tints. — The  great  advantage  which  this  method 
possesses  over  the  one  just  described,  consists  in  imparting  to  the  shade  a 
much  softer  appearance ;  the  limitations  of  the  different  tints  being  imper- 
ceptible. On  the  other  hand,  it  is  considerably  more  difficult,  requiring 
longer  practice,  and  greater  mastery  over  the  movements  of  the  brush  to 
accomplish  it  with  tolerable  precision. 

Let  it  be  proposed  to  shade  by  this  method  the  segment  of  the  hexa- 
gonal pyramid  (fig.  8,  plate  LXXXIV.) 

The  plan  of  this  figure  is  similar  to  that  of  the  prism  (fig.  4,  plate 
LXXXIII.)  Its  position  in  reference  to  the  light  is  also  the  same.  Thus 
the  face  abed  should  receive  a  uniform  flat  tint.  If,  however,  it  be  de- 
sired to  adhere  rigorously  to  the  preceding  rules,  the  tint  may  be  slightly 
deepened  as  it  approaches  the  top  of  the  pyramid,  seeing  that  the  surface 
is  not  quite  parallel  to  the  vertical  plane. 

The  face  b  g  h  c  being  inclined  and  in  the  shade,  should  receive  a  dark 
tint.  The  darkest  part  of  this  tint  is  where  it  meets  the  line  b  c,  and  grad- 
ually becomes  lighter  as  it  approaches  the  line  g  h.  To  produce  this 
eifect,  apply  a  narrow  strip  of  tint  to  the  side  b  c  (fig.  6),  and  then,  quali- 
fying the  tint  in  the  brush  with  a  little  water,  join  another  strip  to  this, 
and  finally,  by  means  of  another  brush  moistened  with  water,  soften  off  this 
second  strip  towards  the  line  1, 1,  which  may  be  taken  as  the  limit  of  the 
first  tint.  This  is  shown  in  fig.  6. 

When  the  first  tint  is  dry,  cover  it  with  a  second,  which  must  be  simi- 
larly treated,  and  should  extend  beyond  the  first  up  to  the  line  2,  2  (fig. 
7).  Proceed  in  this  manner  with  other  tints,  until  the  whole  face  l>  y  h  c 
is  shaded,  as  presented  in  fig.  8. 

In  the  same  way  the  face  e  a  df  is  to  be  covered,  though  with  a  con- 
siderably lighter  tint,  for  the  rays  of  light  happen  to  fall  upon  it  almost 
perpendicularly. 

It  may  be  observed,  that  consistently  to  carry  out  the  rules  we  have 
laid  down,  the  tint  on  these  two  faces  should  be  slightly  graduated  from 
e  a  tofd,  and  from  c  h  to  b  g.  But  this  exactitude  may  be  disregarded 
until  some  proficiency  in  shading  has  been  acquired. 

It  is  now  proposed  to  shade  the  cylinder  (fig.  4)  by  means  of  softened 
tints.  The  boundary  of  each  tint  being  indicated  in  a  manner  precisely 
similar  to  that  shown  by  fig.  5,  plate  LXXXIII.,  the  first  strip  of  tint  must 
cover  the  line  of  extreme  shade  a  b,  and  then  be  softened  off  on  each  side, 
as  shown  in  fig.  13.  Other  and  successively  wider  strips  of  tint  are  to  fol- 
low, and  receive  the  same  treatment  as  the  one  first  put  on.  The  results 
of  this  process  are  shown  in  figs.  2,  3,  and  4. 


SHADING  AND  SHADOWS.  333 

As  this  method  requires  considerable  practice  before  it  can  be  per- 
formed with  much  nicety,  the  learner  need  not  be  discouraged  at  the  fail- 
ure of  his  first  attempts,  but  persevere  in  practising  on  simple  figures  of 
different  sizes. 

If,  after  shading  a  figure  by  the  foregoing  method,  any  very  apparent 
inequalities  present  themselves  in  the  shade,  such  defects  may  be  remedied 
in  some  measure  by  washing  off  redundancies  of  tint  with  a  brush  or  a 
damp  sponge,  and  by  supplying  a  little  color  to  those  parts  which  are  too 
light. 

Dexterity  in  shading  figures  by  softened  tints  will  be  facilitated  in  prac- 
tising upon  large  surfaces ;  this  will  be  the  surest  way  of  overcoming  that 
timidity  and  hesitation  which  usually  accompany  all  first  attempts,  but 
which  must  be  laid  aside  before  much  proficiency  in  shading  can  be  ac- 
quired. 

ELABOKATION  OF  SHADING  AND  SHADOWS. 

Thus  far  the  simplest  primary  rules  for  shading  isolated  objects  have 
been  laid  down,  and  the  easiest  methods  of  carrying  them  into  operation 
explained.  It  is  now  proposed  to  exemplify  these  rules  upon  more  com- 
plex forms,  to  show  where  the  shading  may  be  modified  or  exaggerated, 
to  introduce  additional  rules  more  especially  adapted  for  mechanical  color- 
ing, and  to  offer  some  observations  and  directions  for  effectively  shading 
the  drawing  of  machines  in  their  entirety. 

Whatman's  best  rough-grained  drawing-paper  is  better  adapted  for 
receiving  color  than  any  other.  Of  this  paper,  the  double  elephant  size  is 
preferable,  as  it  possesses  a  peculiar  consistency  and  grain.  A  larger 
paper  is  seldom  required,  and  when  the  drawing  to  be  made  happens  to  be 
small,  a  portion  of  a  double  elephant  sheet  should  be  used. 

The  paper  for  a  colored  drawing  ought  always  to  be  strained  upon  a 
board  with  glue  or  strong  gum.  Before  doing  this,  care  must  be  taken  to 
damp  the  face  of  the  paper  with  a  sponge  well  charged  with  water,  in 
order  to  remove  any  impurities  from  its  surface,  and  as  a  necessary  prepa- 
ration for  the  better  reception  of  the  color.  The  sponge  should  merely 
touch  the  paper  lightly,  and  not  rub  it.  The  whole  of  the  surface  is  to  be 
damped,  that  the  paper  may  be  subjected  to  a  uniform  degree  of  expansion, 
thereby  insuring,  as  it  dries,  a  uniform  degree  of  contraction.  Submitted 
to  this  treatment,  the  sheet  of  paper  will  present,  when  thoroughly  dry,  a 
clean  smooth  surface,  not  only  agreeable  to  work  upon,  but  also  in  the 
best  possible  condition  to  take  the  color. 

The  size  of  the  brushes  to  be  used  will,  of  course,  depend  upon  the 


334:  SHADING  AND   SHADOWS. 

scale  to  which  the  drawing  is  made.  Long  thin  brushes,  however,  should 
be  avoided.  Those  possessing  corpulent  bodies  and  fine  points  are  to  be 
preferred,  as  they  retain  a  greater  quantity  of  color,  and  are  more  manage- 
able. 

During  the  process  of  laying  on  a  flat  tint,  if  the  surface  be  large — 
though  this  is  seldom  the  case  except  in  topographical  drawings — the  draw- 
ing may  be  slightly  inclined,  and  the  brush  well  charged  with  color,  so  that 
the  edge  of  the  tint  may  be  kept  in  a  moist  state  until  the  whole  surface  is 
covered.  In  tinting  a  small  surface,  the  brush  should  never  have  much 
color  in  it,  for  if  it  have,  the  surface  will  unavoidably  present  coarse 
rugged  edges,  and  a  coarse  uneven  appearance  throughout.  A  moderate 
quantity  of  color  in  the  brush,  well  though  expeditiously  rubbed  into  the 
.  paper,  is  the  only  method  of  giving  an  even  close-grained  aspect  to  the 
surface.  In  fact,  for  mechanical  drawings,  there  is  rarely  occasion  for  well 
charging  the  brush  with  color.  The  tint  in  the  brush  may  be  very  dark 
or  very  light,  but  there  should  seldom  be  much  of  it. 

As  an  invariable  rule  let  it  be  remembered,  that  no  tint,  shade,  or 
shadow  is  to  be  passed  over  or  touched  until  it  is  quite  dry. 

In  the  examples  of  shading  which  are  given  in  this  work,  it  may  be 
observed  that  all  objects  with  curved  outlines  have  a  certain  amount  of 
reflected  light  imparted  to  them.  It  is  true  that  all  bodies,  whatever  may 
be  their  form,  are  affected  by  reflected -light ;  but,  with  a  few  exceptions, 
this  light  is  only  appreciable  on  curved  surfaces.  The  judicious  degree 
and  treatment  of  this  light  is  of  considerable  importance  for  the  acquire- 
ment of  an.  effective  style  of  shading. 

All  bodies  in  the  light  reflect  on  those  objects  which  surround  them 
more  or  less  light  according  to  the  situation.  Wherever  light  extends,  re- 
flection follows.  If  an  object  be  isolated,  it  is  still  reached,  by  reflected 
light,  from  the  ground  on  which  it  rests,  or  from  the  air  which  surrounds  it 

In  proportion  to  the  degree  of  polish  or  brightness  in  the  color  of  a 
body,  is  the  amount  of  reflected  light  which  it  spreads  over  adjacent  ob- 
jects, and  also  its  own  susceptibility  of  illumination  under  the  reflection 
from  other  bodies.  A  polished  steam-cylinder,  or  a  white  porcelain  vase, 
receives  and  imparts  more  reflected  light  than  a  rough  casting  or  a  stone 
pitcher. 

Shade,  even  the  most  inconsiderable,  ought  never  to  extend  to  the  out- 
line of  any  smooth  circular  body.  On  a  polished  sphere,  for  instance,  the 
shade  should  be  delicately  softened  off  just  before  it  meets  the  circumfer- 
ence, and  when  the  shading  is  completed,  the  body  color  intended  for  the 
sphere  may  be  carried  on  to  its  outline.  This  will  give  a  transparency  to 


SHADING   AND   SHADOWS.  335 

that  part  of  tlio  sphere  influenced  by  reflected  light,  which  it  could  not 
have  possessed  if  the  shade  tint  had  been  extended  to  its  circumference. 
Very  little  shade  should  be  suffered  to  reach  the  outlines  even  of  rough 
circular  bodies,  lest  the  coloring  look  harsh,  and  present  a  coarse  appear- 
ance quite  at  variance  with  its  natural  aspect.  Shadows  also  become 
lighter  as  they  recede  from  the  bodies  which  cast  them,  owing  to  the  in- 
creasing amount  of  reflection  which  falls  on  them  from  surrounding  ob- 
jects. 

Shadows  appear  to  increase  in  depth  as  their  distance  from  the  spectator 
diminishes.  In  nature  this  increase  is  only  appreciable  at  considerable 
distances.  Even  on  extensive  buildings,  inequalities  in  the  depth  of  the 
shadows  are  hardly  perceptible ;  much  less,  then,  can  any  natural  grada- 
tion present  itself  in  the  shadows  on  a  machine,  which,  supposing  it  to  be 
of  the  largest  construction,  is  confined  to  a  comparatively  small  space.  It 
is  most  important,  however,  for  the  effective  representation  of  machinery, 
that  the  variation  in  the  distance  of  each  part  of  a  machine  from  the  spec- 
tator should  at  once  strike  the  eye ;  and  an  exaggeration  in  expressing  the 
varying  depths  of  the  shadows  is  one  means  of  effecting  that  object.  The 
shadows  on  the  nearest  and  most  prominent  parts  of  a  machine  should  be 
made  as  dark  as  color  can  render  them,  the  colorist  being  thus  enabled  to 
exhibit  a  marked  difference  in  the  shadows  on  the  other  parts  of  the  ma- 
chine as  they  recede  from  the  eye.  The  same  direction  is  applicable  in 
reference  to  shades.  The  shade  on  a  cylinder,  for  instance,  situated  near 
the  spectator,  ought  to  be  darker  than  on  one  more  remote  ;  in  fact,  the 
gradation  of  depth  for  the  shades  follows  that  which  depicts  the  shadows. 
As  a  general  rule,  the  color  on  a  machine,  no  matter  what  it  may  be  in- 
tended to  represent,  should  become  lighter  as  the  parts  on  which  it  is 
placed  recede  from  the  eye. 

Plates  LXXXY.  and  LXXXYI.  present  some  very  good  examples  of 
finished  shading. 

Plate  LXXXV,  represents,  both  in  elevation  and  plan,  different  solids 
variously  penetrated  and  intersected.  The  rules  for  the  projection  of  these 
solids  have  been  given  under  the  head  of  Geometrical  Projection,  and  illus- 
trated in  plates  VI.,  VII.,  VIIL,  IX.,  X.  They  are  selected  with  a  view 
of  exhibiting  those  cases  which  are  of  most  frequent  occurrence,  and  at  the 
same  time  elucidating  the  general  principles  of  shading. 

Plate  LXXXVI.  presents  examples  of  shading  and  shadow. 

Fig.  1  presents  a  hexagonal  prism  surmounted  by  a  fillet.  The  most 
noticeable  part  of  this  figure  is  the  shadow  of  the  prism  in  the  plan  view. 
It  presents  a  good  example  of  the  graduated  expression  which  should  be 


336  SHADING  AND  SHADOWS. 

given  to  all  shadows  cast  upon  plain  surfaces.  Its  two  extremities  are  re- 
markably different  in  their  tone.  As  the  shadow  nears  the  prism,  it  in- 
creases rapidly  in  depth  ;  on  the  contrary,  as  it  approaches  the  other  end, 
it  assumes  a  comparatively  light  appearance.  This  difference  is  doubt- 
lessly a  great  exaggeration  upon  what  it  would1  naturally  display.  Any 
modification  of  it,  however,  in  the  representation  would  destroy  the  best 
effect  of  the  shadow. 

The  direction  which  the  shades  and  shadows  take  in  all  the  plans  of 
the  figures  in  this  plate,  is  from  the  left  hand  lower  corner.  This  is  rigor- 
ously correct,  supposing  the  objects  to  remain  stationary,  whilst  the  spec- 
tator views  them  in  both  a  vertical  and  horizontal  position.  Nevertheless, 
to  many,  this  upward  direction  given  to  the  shadows  has  an  awkward 
appearance,  and,  perhaps,  in  the  plan  of  an  entire  machine,  the  shadows 
may  look  better  if  their  direction  coincide  with  that  which  is  given  to 
them  in  the  elevation.  If,  however,  the  shadows  be  correctly  projected, 
their  direction  is  an  arbitrary  matter,  and  may  be  left  to  the  taste  of  the 
draughtsman. 

Figs.  2,  3,  and  6  exemplify  the  complex  appearance  of  shade  and 
shadow  presented  on  concave  surfaces.  It  is  worthy  of  particular  notice, 
that  the  shadow  on  a  concave  surface  is  darkest  towards  its  outline,  and 
becomes  lighter  as  it  nears  the  edge  of  the  object.  Reflection  from  that 
part  of  the  surface  on  which  the  light  falls  most  powerfully  causes  this 
gradual  diminution  in  the  depth  of  the  shadow,  the  greatest  amount  of  re- 
flection being  opposite  the  greatest  amount  of  light. 

It  may  be  as  well  to  remark  here,  that  no  brilliant  or  extreme  light 
should  be  left  on  concave  surfaces,  as  such  lights  would  tend  to  render  it 
doubtful  at  first  sight  whether  the  objects  represented  were  concave  or 
convex.  After  the  body-color — which  shall  be  treated  in  a  subsequent 
section — has  been  put  on,  a  faint  wash  should  be  passed  very  lightly  over 
the  whole  concavity.  This  will  not  only  modify  and  subdue  the  light,  but 
tend  to  .soften  any  asperities  in  the  tinting,  which  are  more  unsightly  on  a 
concave  surface  fhan  on  any  other. 

The  lightest  part  of  a  sphere  (fig.  4)  is  confined  to  a  mere  point,  around 
which  the  shade  commences  and  gradually  increases  as  it  recedes.  This 
point  is  not  indicated  on  the  figure  referred  to,  because  the  shade  tint  on 
a  sphere  ought  not  to  be  spread  over  a  greater  portion  of  its  surface  than 
is  shown  there.  The  very  delicate  and  hardly,  perceptible  progression  of 
the  shade  in  the  immediate  vicinity  of  the  light  point  should  be  effected 
by  means  of  the  body-color  of  the  sphere.  If,  for  instance,  the  material 
of  which  the  sphere  is  composed  be  brass,  the  body-color  itself  should  be 


SHADING  AND  SHADOWS.  337 

lightened  as  it  nears  the  light  point.  In  like  manner  all  polished  or  light- 
colored  curved  surfaces  should  be  treated  ;  the  part  bordering  upon  the  ex- 
treme light  being  covered  with  a  tint  of  body-color  somewhat  fainter  than 
that  used  for  the  flat  surfaces.  Again,  if  the  sphere  be  of  cast-iron,  then  the 
ordinary  body-color  should  be  deepened  from  the  light  point  until  it  meets 
the  shade  tint,  over  which  it  is  to  be  spread  uniformly.  Any  curved  un- 
polished surface  is  to  be  thus  treated  ;  the  body-color  should  be  gradually 
deepened  as  it  recedes  from  that  part  of  the  surface  most  exposed  to  the 
light.  Considerable  management  is  necessary  in  order  to  shade  a  sphere 
effectively.  The  best  way  is  to  put  on  two  or  three  softened-off  tints  in 
the  form  of  crescents  converging  towards  the  light  point,  the  first  one 
being  carried  over  the  point  of  deepest  shade. 

A  ring  (fig.  5)  is  a  difficult  object  to  shade.  To  change  with  accurate 
and  effective  gradation  the  shade  from  the  inside  to  the  outside  of  the  ring, 
to  leave  with  regularity  a  line  of  light  upon  its  surface,  and  to  project  its 
shadow  with  precision,  require  a  degree  of  attention  and  care  in  their  exe- 
cution greater,  perhaps,  than  the  shade  and  shadow  of  any  other  simple 
figure.  The  learner,  therefore,  should  practise  the  shading  of  this  figure, 
as  he  will  seldom  meet  with  one  presenting  greater  difficulties. 

Figs.  7  and  8  show  the  peculiarities  of  the  shadows  cast  by  a  conical 
form  on  a  sphere  or  cylinder.  The  following  fact  should  be  well  noted  in 
the  memory  : — That  the  depth  of  a  shadow  on  any  object  is  in  proportion 
to  the  degree  of  light  which  it  encounters  on  the  surface  of  that  object. 
In  these  figures  very  apt  illustrations  of  this  fact  may  be  remarked.  It 
will  be  seen  by  referring  to  the  plan  (fig.  7),  that  the  shadow  of  the  apex 
of  the  cone  happens  to  fait  upon  the  lightest  point  of  the  sphere,  and  is, 
therefore,  the  darkest  part  of  the  shadow.  So  also  the  deepest  portion  of 
the  shadow  of  the  cone  on  the  cylinder  in  the  plan  (fig.  8)  is  exactly  where 
it  coincides  with  the  line  of  extreme  light.  Flat  surfaces  are  ^similarly 
affected,  the  shadows  thrown  on  them  being  less  darkly  expressed,  accord- 
ing as  their  inclination  to  the  plane  of  projection  increases.  The  body- 
color  on  a  flat  surface  should,  on  the  contrary,  increase  in  depth  as  the  sur- 
face becomes  more  inclined  to  this  plane. 

Another  notable  fact  is  exemplified  by  these  figures  : — that  reflected 
light  is  incident  to  shadows  as  well  as  to  shades.  This  is  very  observable 
where  the  shadow  of  the  cone  falls  upon  the  cylinder.  It  may  likewise 
be  remarked,  though  to  a  less  extent,  on  other  parts  of  these  figures.  The 
reflected  light  on  the  cone  from  the  sphere  or  cylinder  is  also  worthy  of 
observation.  This  light  adds  greatly  to  the  effect  of  the  shadows,  and 
22 


338  SHADING  AND  SHADOWS. 

indeed  to  the  appearance  of  the  objects  themselves.  Altogether,  these 
figures  offer  admirable  scope  for  study  and  practice. 

The  concentration  within  a  small  space  of  nearly  all  the  peculiarities 
and  effects  of  light,  shade,  and  shadow,  may  be  seen  on  plate  LXXXVIL, 
in  the  examples  of  screws  there  given. 

The  parts  of  a  highly-finished  colored  drawing  of  a  machine  are  always 
affected  by  a  certain  degree  of  indefinableness  in  their  outline. 

Notwithstanding  the  most  careful  exertions  of  the  colorist  to  keep 
every  feature  of  a  machine  clear  and  distinct,  some  amount  of  uncertainty, 
resulting  unavoidably  from  the  proximity  and  natural  blending  of  the  dif- 
ferent parts,  will  pervade  the  lines  which  separate  its  component  members. 
For  practical  working  purposes,  therefore,  a  completely  colored  drawing 
of  a  machine  is  unsuitable.  On  the  other  hand,  a  mere  outline,  although, 
perhaps,  intelligible  enough  to  those  who  are  familiarly  acquainted  with 
the  machine  delineated,  has  an  undecided  appearance.  As  complete 
coloring  renders  it  difficult  for  the  eye  to  separate  the  various  parts  of  a 
machine,  owing  to  an  apparently  too  intimate  relationship  between  them  ; 
a  line  drawing,  on  the  contrary,  perplexes  the  eye  to  discover  any  relation 
between  them  at  all,  or  to  settle  promptly  their  configuration.  The  eye 
involuntarily  asks  the  question,  is  that  part  round  or  square,  or  is  it  even 
a  distinct  part  of  the  machine  at  all  ?  As  a  means  of  avoiding  the  inde- 
finiteness  presented  by  the  outline  in  the  former  case,  and  the  want  of 
adequate  coherence  and  doubtfulness  in  the  form  of  the  different  parts 
amenable  to  the  latter,  recourse  is  not  unfrequently  had  to  a  kind  of  semi- 
coloring,  or  rather  mere  shading  of  the  parts  of  a  machine.  Exemplifi- 
cations of  this  practical  style  of  representing  machines  may  be  observed 
in  plates  XL.  to  XLIII.  inclusive,  "  Drawing  of  Machinery,"  pages 
204  and  206.  Every  figure  looks  complete  without  elaboration,  and  is 
clearly  Delineated  without  degenerating  into  the  bareness  of  a  mere  skele- 
ton. Outlines  and  forms  are  at  once  apprehended,  and  every  member  of 
the  machine  is  adjusted  without  hesitation  to  its  proper  place. 

In  such  drawings  shading  only  is  allowed,  and  therefore  but  slight 
scope  is  permitted  for  imparting  effects  ;  and  it  is  advisable  to  follow  a 
direction  previously  given,  and  to  modify  the  color  on  every  part  accord- 
ing to  its  distance  from  the  eye.  It  may  be  as  well  also,  for  the  purpose 
of  maintaining  harmony  in  the  coloring,  and  of  equalizing  its  appearance, 
to  color  less  darkly  large  shades  than  small  ones,  although  they  may  be' 
situated  at  an  equal  distance  from  the  eye.  No  very  dark  shading  is  per- 
missible on  this  species  of  drawing ;  indeed  the  tint  should  be  very  con- 
siderably lighter  than  on  finished  colored  drawings.  Besides  presenting 


SHADING  AND  SHADOWS.  339 

too  violent  a  contrast  between  the  parts  colored  and  those  without  any 
color  at  all,  dark  shading  would  produce,  in  some  measure,  the  indistinct- 
ness which  is  objectionable  in  completely  tinted  drawings. 

At  page  383,  Plate  XCYIII.  is  a  photograph  from  a  finished  drawing 
of  the  engine  and  a  boiler  of  the  steamer  Pacific.  Every  shadow  is  care- 
fully projected,  every  detail  elaborated,  and  the  execution  perfect;  it  may 
serve  as  a  model  of  its  class,  not  only  for  its  accuracy  and  distinctness  of 
detail,  but  also  for  its  vigor  as  a  picture.  It  is  seldom  that  so  much  labor 
is  devoted  to  a  mechanical  drawing,  but  the  result  is  very  satisfactory  to 
the  designer. 

FINISHED     COLORING. 

The  coloring  of  drawings  representing  machinery  requires  a  special 
study,  the  process  of  its  development  being,  in  many  essentials,  very  differ- 
ent from  that  pursued  in  the  artistic  expression  of  other  objects. 

The  parts  of  a  machine  being  usually  constructed  with  mathematical 
accuracy,  and  always  presenting  a  well-defined  rigid  outline,  the  same 
unmistakable  definiteness  should  be  maintained  in  any  attempt  to  picture 
such  an  object  on  paper.  There  should  be  no  "blending"  of  different 
colors,  no  doubtful  finish  to  a  tint,  no  softening  off  into  the  imaginative  ; 
every  part  should  present  at  once  to  the  eye  its  form  and  position ;  should, 
in  fact,  supply  the  place  of  a  model  of  the  machine.  So  important  a  fea- 
ture is  this  in  mechanical  coloring,  that  when  correct  shadows  would  mate- 
rially obscure  any  part  of  a  machine,  they  should  either  be  entirely  sup- 
pressed, or,  when  such  an  omission  would  be  very  striking,  so  modified  as 
to  lessen  as  much  as  possible  the  obscurity  thus  produced. 

The  color  of  cast-iron  fresh  from  the  foundry  is  commonly  a  very  dark 
bluish  black,  having  blended  with  it  an  almost  imperceptible  brownish- 
green  tint  or  cast.  To  represent  the  casting  on  paper  to  the  best  advan- 
tage, the  following  colors  should  be  employed : — Indian  ink  and  indigo, 
with  a  very  slight  admixture  of  lake.  This  last  ingredient  is  necessary ; 
for  Indian  ink,  being  actually  only  a  very  dark  brown,  it  would,  in  con- 
junction merely  with  a  blue,  impart  too  green  a  cast  to  the  tint  sought  to 
be  realized. 

Great  care  should  be  taken  in  mixing  these  colors.  First,  the  lake — 
crimson  is  preferable — should  be  rubbed  on  the  pallet ;  about  half-a-dozen 
turns  of  the  hand  are  sufficient,  as  too  much  of  this  color  would  impart  a 
rusty  appearance  to  the  desired  tint.  The  indigo  may  then  be  added, 
and  lastly  the  Indian  ink.  The  quantity  of  lake  being  very  inconsiderable, 
about  two-thirds  of  the  mixture  should  be  composed  of  Indian  ink,  and 


34K)  SHADING  AND   SHADOWS. 

the  remaining  third  of  indigo.  This  proportion,  however,  will  be  best 
ascertained  by  occasionally  trying  the  tint  on  a  scrap  of  drawing-paper 
during  the  process  of  mixing.  "When  the  tint  appears  to  have  approxi- 
mated as  near  as  possible,  according  to  the  colorist's  judgment,  to  the  tint 
described  above,  its  ingredients  should  be  well  mixed  together  with  the 
brush.  The  more  intimate  this  intermixture  of  the  colors  can  be  rendered, 
the  better  ;  for  if  any  considerable  number  of  particles  of  the  same  color 
remain  together,  the  tint,  when  essayed,  will  present  a  streaky,  semi-party- 
colored  appearance.  The  tint  being  thus  prepared,  should  be  left  for  a  short 
time  untouched,  so  as  to  allow  the  grosser  particles  of  color  to  settle  at  the 
bottom  of  the  saucer.  Eo  fear  need  be  entertained  of  getting  the  tint  too 
dark,  or  of  mixing  too  much  ;  on  the  contrary,  it  is  better  to  compound  a 
considerable  quantity  and  very  dark  in  one  saucer,  and  then  gently  pour 
a  little  into  one  or  two  others,  in  which,  with  varying  quantities  of  water, 
different  gradations  of  tint  may  be  produced.  The  tint  left  in  the  first 
pallet  should  be  preserved  for  shading  or  for  shadows,  and  when  it  has  be- 
come dry,  should  by  no  means  be  discarded,  as  it  will  always  be  service- 
able, and  indeed  preferable  for  imparting  the  lesser  dark  effects  to  various 
parts  of  the  drawing. 

With  one  or  two  exceptions,  which  will  be  pointed  out  later,  this  tint, 
variously  modified,  is  the  only  one  to  be  employed  for  the  representation 
of  cast  iron.  It  is  adapted  as  well  for  expressing  the  shades  and  shadows 
as  for  depicting  the  body-color.  If  the  shades  and  shadows  be  indicated 
by  Indian  ink  alone,  the  small  amount  of  "brilliancy"  which  cast  iron 
naturally  enjoys  will  disappear  wherever  covered  with  Indian  ink,  and 
even  the  effect  of  the  body-color  will  be  very  sensibly  diminished. 

The  first  parts  of  the  drawing  of  a  machine  which  it  is  usually  most 
judicious  to  color  are  those  of  a  circular  form — cylinders,  the  more  im- 
portant shafts,  &c.  The  rods  and  smaller  shafts,  especially  where  they 
cross  other  parts  of  the  machine,  may  be  left  until  the  other  work  is 
finished. 

Taking  for  granted  that  the  learner  has  practised  the  art  of  shading 
according  to  the  simple  methods  previously  described,  and,  therefore,  that 
he  is  somewhat  acquainted  with  the  use  of  the  brush,  let  him  now  proceed 
to  color  a  circular  casting,  it  being  with  cast  iron  only  that  we  have  to  do 
at  present. 

Imagine  this  casting  to  be  a  large  cylinder.  First  draw  two  faint 
pencil  lines,  to  indicate  the  extremes  of  light  and  shade  on  its  surface. 
Pass  the  brush,  moderately  full  of  the  darkest  tint,  down  the  line  of  deep- 
est shade,  spreading  the  color  more  or  less  on  either  side,  according  to  the 


SHADING  AND  SHADOWS.  341 

diameter  of  the  cylinder  ;  then,  if  possible,  before  this  layer  of  tint  is  dry, 
towards  the  line  of  extreme  light,  beginning  at  the  top,  and  encroaching 
slightly  over  the  edge  of  the  first  tint,  lay  on  another  not  quite  so  dark, 
but  about  double  its  width.  It  may  be  observed,  that  it  is  not  very  essen- 
tial to  put  on  the  second  tint  before  the  first  is  dry,  for  the  latter  should 
be  so  dark  and  thick,  that  its  edges  may  be  easily  softened  at  any  time. 
Whilst  this  second  tint  is  still  wet,  with  a  much  lighter  color  in  the  brush, 
proceed  in  the  same  manner  with  a  third  tint,  and  so  on,  until  the  line  of 
extreme  light  is  nearly  attained.  Repeat  this  process  on  the  other  side  of 
the  first  tint,  approaching  the  outline  of  the  cylinder  with  a  very  faint 
wash,  so  as  to  represent  the  reflected  light  which  progressively  modifies 
the  shade  as  it  nears  that  line.  Then  let  a  darkish  narrow  strip  of  tint 
meet,  and  pass  along  the  outline  of  the  cylinder  on  the  other  side  of  the 
extreme  line  of  light,  after  which  gradually  fainter  tints  should  follow, 
treated  in  a  manner  similar  to  that  which  has  been  already  described,  and 
becoming  almost  imperceptible  just  before  arriving  at  the  line  of  light. 

This  is  a  very  expeditious  way  of  shading  a  cylinder ;  but  even  to  the 
most  experienced  colorist,  it  is  not  possible,  by  the  above-described  means 
alone,  to  impart  a  sufficient  degree  of  well-regulated  rotundity  to  the  ap- 
pearance of  such  an  object.  Superfluities  and  deficiencies  of  color  will 
appear  here  and  there.  It  will  be  necessary,  therefore,  to  equalize  to  some 
extent,  by  a  species  of  gross  stippling,  the  disparities  which  present  them- 
selves. This  is  done  by  spreading  a  little  color  over  the  parts  where  it  is 
deficient,  and  then  passing  very  lightly  over  nearly  the  whole  width  of  the 
shade,  with  the  brush  supplied  with  a  very  light  wash.  This  process  may 
be  repeated  to  suit  the  degree  of  finish  whichr  it  is  desired  to  give  the 
drawing.  In  the  same  manner  the  shading  of  all  curved  surfaces  is  to  be 
treated. 

Recourse  is  often  had  to  what  is  called  "  washing  "  or  "  sponging,"  in 
order  to  impart  softness  and  circularity  to  certain  forms.  Beyond  a  very 
limited  extent,  this  is  a  most  injudicious  system.  It  robs  the  shade  of  all 
the  lightest  and  most  brilliant  particles  of  color,  the  natural  position  of 
which  is  on  the  surface ;  it  destroys  that  "  crisp  "  freshness,  so  essential 
towards  the  beautiful  appearance  of  all  coloring ;  and,  what  is  still  worse, 
spreads  a  dirty  appearance  not  only  over  the  whole  surface  of  the  coloring, 
but  more  or  less  on  all  the  paper  which  surrounds  it.  Sponging  should 
never  be  adopted,  and  if  a  slight  washing  with  the  brush  be.  sometimes 
attempted,  it  should  be  done  very  lightly,  and,  except  on  rare  occasions, 
not  allowed  to  pass  beyond  those  parts  of  the  drawing  covered  with  color, 
otherwise  that  sharp  cleanly  appearance,  which  so  enhances  the  effect  of  a 


3i2  SHADING  AND  SHADOWS. 

colored  drawing,  will  be  lost.  Let  it,  then,  be  remembered,  that  the  less 
the  color,  whether  as  a  shade,  shadow,  or  tint  of  any  kind,  is  touched  after 
it  has  reached  the  paper,  the  better.  The  system  of  shading  by  numerous 
tints  laid  one  over  the  other — a  system  which  almost  universally  prevails 
— is  no  doubt  a  very  easy,  and,  therefore,  advantageous  one  for  the  initia- 
tion of  beginners  into  a  dexterous  use  of  the  brush  and  the  grosser  mysteries 
of  coloring ;  but  no  highly  effective  mechanical  drawing  can  be  produced 
in  this  manner. 

The  principal  shadows  are  the  next  parts  of  the  coloring  which  will  now 
claim  attention.  The  outline  of  any  shadow  being  drawn  in  pencil,  along 
its  inner  line — the  line  which  forms  a  portion  of  the  figure  of  the  object 
whose  shadow  is  to  be  represented — lay  on  a  strip  of  the  darkest  tint,  wide 
or  narrow,  according  to  the  width  of  the  shadow,  and  then,  before  it  is 
dry,  soften  off  its  outer  edge.  This  may  be  repeated  as  often  as  the  taste 
of  the  colorist  may  dictate,  but  the  color  should  not  spread  itself  over 
much  more  than  half  the  space  occupied  by  the  shadow.  These  prelimi- 
nary touches  will  add  to  the  intensity  of  the  proposed  shadow,  and  neutralize 
a  certain  harshness  of  appearance  inevitable  to  all  shadows  made  equally 
dark  throughout.  The  effect  they  give  to  the  drawing  is  very  pleasing, 
and  is,  moreover,  quite  natural,  for,  as  previously  explained,  the  greatest 
depth  of  a  shadow  is  invariably  that  part  of  it  immediately  contiguous  to 
the  object  shadowed  forth. 

The  representation  of  the  casting  is  now  to  be  completed  by  laying  on 
the  body-color.  This  might  be  done  by  a  single  wash  of  tint  if  the  ap- 
pearance of  cast  iron  were  as  light  as  it  is  usually  depicted ;  but  its  natural 
color  being,  on  the  contrary,  very  opaque  and  heavy,  two  and  sometimes 
three  washes  are  necessary,  the  first  tint  being  rather  darker  than  those 
which  follow.  Each  tint  should  pass  over  the  shades  and  shadows  when 
they  occur,  care  being  taken  to  manoeuvre  the  brush  at  such  parts  very 
lightly. 

The  most  conspicuous  fault  observable  in  the  generality  of  colored 
mechanical  drawings  is  a  deficiency  in  the  depth  of  the  tints  employed. 
There  appears  to  exist  an  undefinable  fear  of  transferring  to  paper  the 
naturally  dark  appearance  of  iron ;  the  result  is  the  production  of  tame, 
ineffective  representations,  which,  instead  of  looking  as  they  should,  like 
models  of  iron  machines,  present  mere  faint  shadows  of  such  objects,  or, 
at  best,  machinery  constructed  of  some  unknown,  light,  and  rather  dirty 
materials. 

The  sectional  surfaces  -of  cast-iron  are  to  be  indicated  by  one  light  tint 
of  indigo. 


SHADING  AND  SHADOWS.  343 

The  next  most  extensive  and  important  component  used  in  the  manu- 
facture of  machines  is  wrought  iron.  Precisely  the  same  colors  are  to  be 
employed  to  represent  this  material  as  have  been  pointed  out  for  cast-iron. 
The  difference  in  the  appearance  of  these  metals  is  produced  by  altering 
the  proportion  of  the  two  principal  colors, — Indian  ink  and  indigo.  These 
ingredients  should  be  mixed,  carefully  and  well  mixed,  in  about  equal  pro- 
portions, a  very  small  quantity  of  crimson  lake  being  first  rubbed  in  the 
saucer. 

The  same  methods  of  shading  and  of  laying  on  the  shadows  prescribed 
for  cast-iron  are  to  be  adopted  in  the  case  of  wrought-iron,  keeping,  how- 
ever, all  parts  of  the  latter  lighter,  particularly  the  body-color.  The  direct 
and  reflected  lights  must  also  present  themselves  more  distinctly,  and  to  a 
much  greater  extent.  Polished  and  semi-polished  surfaces  invariably 
afford  greater  contrasts  of  light  and  shade  than  other  surfaces.  The  steps 
or  rather  glidings  from  one  extreme  to  the  other  are,  moreover,  softer  and 
more  delicately  graduated,  and,  therefore,  greater  care  is  requisite  in  repre- 
senting them  on  paper.  These  remarks  are  very  effectively  illustrated  by 
the  fragments  of  large  screws  shown  on  pi.  LXXXYII.  and  also  by  the 
photograph  of  the  steamer  Pacific,  Plate  XCVIII. 

For  the  parts  of  wrought  iron  in  section  a  light  tint  of  Prussian  blue  is 
most  suitable.  This  is  the  only  service  for  which  Prussian  blue  can  pro- 
perly be  made  available  in  coloring  drawings  of  machinery.  In  conjunc- 
tion with  Indian  ink  or  indigo  its  inherent  brightness  entirely  disappears  ; 
an  ill-assorted  union  with  the  former  producing  a  dirty  color,  in  appear- 
ance not  unlike  that  presented  by  the  surface  of  a  stagnant  pool ;  and 
with  the  latter  creating  a  tint  bearing  a  striking  resemblance  to  soiled 
glass.  For  mechanical  drawings,  then,  this  color  must  never  be  used  in 
combination. 

Brass,  except  in  small  quantities,  seldom  makes  its  appearance  in 
machinery.  This  is  fortunate  for  the  colorist,  as  there  is  no  metal  more 
difficult  to  represent  than  brass.  The  body-tint  is  composed  either  of 
gamboge  and  burnt  sienna,  or  gamboge  and  crimson-lake  ;  the  shading 
and  shadows  being  best  expressed  by  burnt  umber. 

The  most  delicate  and  careful  treatment  is  needed  in  making  use  of 
these  colors  ;  for,  when  on  the  paper,  they  are  all  of  them  very  soft,  and 
therefore  highly  sensitive  to  every  touch  of  the  brush.  For  this  reason 
the  shadows  are  best  put  on  after  the  body-color,  otherwise  their  edges 
will  inevitably  present  a  smeary,  indefinite  appearance. 

For  representing  brass  and  copper,  the  method  of  coloring  we  have 
described  in  this  section  is  particularly  suitable.  To  attempt  the  produc- 


3M  SHADING   AND   SHADOWS. 

tion  of  a  shade  with  burnt  umber,  by  means  of  a  succession  of  tints,  would 
merely  realize  a  complicated  smear.  We  find,  therefore,  that  the  shades 
and  shadows  of  brass  are  usually  represented  by  Indian  ink ;  but  as  gam- 
boge almost  invariably  enters  as  an  ingredient  into  the  body-color  of  brass, 
the  result  is  that  the  bright  gamboge  over  the  brown-black  Indian  ink  ex- 
hibits a  species  of  green,  to  which  we  cannot  find  any  thing  comparable, 
but  which  commonly  has  a  very  unpleasant  effect  to  the  eye. 

In  shading  circular  surfaces  great  management  is  requisite.  "  Wash- 
ing "  is  here  entirely  out  of  the  question,  for  even  the  necessary  softening 
off  with  the  brush  is  attended  with  much  difiiculty.  The  brush  should  not 
pass  heavily  or  often  over  the  shade  tint,  lest  unseemly  deficiencies  and 
streaks  of  color  present  themselves  here  and  there,  which  prove  rather 
difficult  blemishes  to  repair.  The  utmost  care  and  experience,  neverthe- 
less, cannot  wholly  insure  the  colorist  against  the  perplexities  of  such 
partial  failures.  The  only  way  to  manage  these  defects  is  by  delicate 
stippling ;  suiting  the  depth  of  tint  to  the  various  degrees  of  shade 
affected,  and  then  passing  a  soft  brush,  filled  moderately  with  dark  body- 
color,  very  lightly  over  the  whole  shade. 

A  light  tint  of  gamboge  is  to  be  used  for  the  sections  of  brass. 

The  directions  which  we  have  given  for  the  most  advantageous  treat- 
ment of  the  colors  representing  brass,  are  equally  applicable  to  those  which 
exhibit  the  nearest  approach  to  copper,  the  colors  to  be  used  for  this  metal 
opposing  nearly  an  equal  amount  of  difiiculty  in  their  management.  A 
mixture  of  orange  chrome  and  lake,  or  red-lead  and  lake,  best  represent 
this  metal ;  its  shades  and  shadows  being  indicated  by  sepia,  whilst  its 
sectioning  is  shown  by  a  light  tint  of  orange  chrome. 

Such  are  the  colors,  and  such  is  the  manner  of  treating  them,  em- 
ployed for  depicting  on  paper  each  of  the  principal  metals  used  in  ma- 
chinery. 

Having  explained  in  detail  the  tinting  of  machinery  in  reference  both 
to  its  shading  and  body-color,  we  propose  to  complete  our  remarks  on 
mechanical  coloring  with  a  few  suggestions  for  imparting  some  peculiar 
effects  to  the  representations  of  masses  of  machinery. 

We  have  already  noticed  that  the  shades  and  shadows  of  a  machine 
are  modified  in  intensity  as  their  distance  from  the  eye  increases.  Its 
body-color  should  be  treated  in  a  similar  manner,  becoming  lighter  and 
less  bright  as  the  parts  of  the  machine  which  it  covers  recede  from  the 
spectator. 

When  the  large  circular  members  of  a  machine  have  been  shaded,  the 


SHADING  AND  SHADOWS.  345 

shadows,  and  even  the  body-color  on  those  parts  furthest  removed  from 
the  eye,  are  to  follow,  and  the  proportion  of  Indian  ink  in .  the  tint  used 
should  increase  as  the  part  to  be  colored  becomes  more  remote.  A  little 
washing,  moreover,  of  the  most  distant  parts  is  allowable,  as  it  gives  a 
pleasing  appearance  of  atmospheric  remoteness,  or  depth,  to  the  color  thus 
treated. 

The  amount  of  light  and  reflection  on  the  members  of  a  machine  should 
diminish  in  intensity  as  the  distance  of  such  objects  from  the  spectator  in- 
creases. As  it  is  necessary,  for  effect,  to  render,  on  those  parts  of  a  m'a- 
chine  nearest  the  eye,  the  contrast  of  light  and  shade  as  intense  as  possible, 
so,  for  the  same  object,  the  light  and  shade  on  the  remotest  parts  should 
be  subdued  and  blended  according  to  the  extent  or  size  of  the  machine. 

A  means  of  adding  considerably  to  the  definiteness  of  a  colored  me- 
chanical drawing,  and  of  promoting,  in  a  remarkable  degree,  its  effective 
appearance,  is  obtained  by  leaving  a  very  narrow  margin  of  light  on  the 
edges  of  all  surfaces,  no  matter  what  may  be  the  angles  which  they  may 
form  with  the  surfaces  that  join  them.  This  should  be  done  invariably  ; 
we  do  not  even  except  those  edges  which  happen  to  have  shadows  falling 
on  them  ;  in  such  cases,  however,  this  margin,  instead  of  being  left  quite 
white,  which  would  have  a  harsh  appearance,  may  be  slightly  subdued. 
The  difficulty  of  achieving  this  effect,  of  imparting  a  clear,  regular,  un- 
broken appearance  to  these  lines  of  light,  seems  very  formidable,  and, 
indeed,  nearly  insuperable.  The  hand  of  the  colorist  may  be  as  steady 
and  confident  as  a  hand  can  be,  and  yet  fail  to  guide  the  brush,  at  an 
almost  inappreciable  distance  from  a  straight  or  circular  line,  with  that 
precision  and  sharpness  so  requisite  for  the  accurate  delineation  of  this 
beautiful  effect.  We  shall,  however,  explain  a  novel  and  effective  method 
of  arriving  at  this  most  desirable  result. 

Suppose  the  object  about  to  receive  the  color  to  be  the  elevation  of  a 
long  flat  rod  or  lever,  on  the  edge  of  which  a  line  of  light  is  to  be  left. 
Fill  the  drawing  pen  as  full  as  it  will  conveniently  hold  with  tint  destined 
to  cover  the  rod  or  lever,  and  draw  a  broad  line  just  within,  but  not  touch- 
ing, the  edge  of  the  lever  exposed  to  the  light.  As  it  is  essential  for  the 
successful  accomplishment  of  the  desired  effect  that  this  line  of  color  should 
not  dry,  even  partially,  until  the  tint  on  the  whole  side  of  the  lever  lias 
been  put  on,  it  will  be  as  well  to  draw  the  pen  again  very  lightly  over 
the  same  part,  so  that  the  line  may  retain  as  much  tint  as  possible.  Im- 
mediately this  has  been  done,  the  brash,  properly  filled  with  the  same 
tint,  is  to  pass  along  and  join  the  inner  edge  of  this  narrow  strip  of  color, 
and  the  whole  surface  of  the  lever  filled  in.  Thus  a  distinct  and  regular 


346  SHADING  AND  SHADOWS. 

line  of  light  is  obtained,  and,  in  fact,  the  lever,  or  whatever  else  the  ob- 
ject may  be,  covered  in  a  shorter  time  than  usual.  A  still  more  expedi- 
tious way  of  coloring  such  surfaces  is  to  draw  a  second  line  of  color  along 
and  joining  the  opposite  edge  of  the  lever  or  other  object,  and  then  expe- 
ditiously  to  fill  in  the  intermediate  space  between  the  two  wet  lines,  by 
means  of  the  brush.  In  this  manner  a  clear  uniform  outline  to  the  tint  is 
obtained,  which  could  not  be  effected  in  any  other  way.  As  celerity-  in 
the  movements  of  the  colorist  is  very  necessary  to  carry  out  properly  this 
method  of  leaving  a  light  edge  to  the  boundaries  of  flat  surfaces,  and  as 
confidence  in  possessing  the  requisite  ability  to  perform  it  must  precede 
success,  a  little  practice  is  desirable  before  essaying  it  on  any  drawing 
of  importance.  The  blades  of  the  drawing  pen  must  not  be  sharp,  and  the 
pen  should  be  used  with  great  precaution  and  delicate  lightness,  otherwise 
the  blades  will  cut  more  or  less  the  paper  and  leave  their  course  visible — 
an  unsightly  betrayal  of  the  mechanical  means  employed  to  obtain  such 
regularity  in  the  coloring.  Flat  circular  surfaces  may  be  treated  in  the 
same  manner,  by  using  the  pen-compass  in  place  of  the  drawing  pen. 
When  such  surfaces  are  rather  extensive,  it  will  be  judicious  to  color  them 
in  halves,  or  in  quadrantal  spaces,  taking  great  care,  when  joining  the 
parts  together,  that  they  may  overlap  or  fall  short  of  each  other  as  little 
as  possible.  The  appearance  of  these  junctions  may  be  obliterated  by 
slightly  washing  them,  or  by  going  over  the  whole  surface  with  a  very 
light  tint,  and,  in  passing,  gently  rubbing  the  seams  with  the  brush.  By 
similar  means  the  line  of  light  on  a  cylinder,  shaft,  or  other  circular  body, 
may  be  beautifully  expressed.  To  indicate  this  light  with  perfect  regu- 
larity is  highly  important,  for  if  a  strict  uniformity  be  not  maintained 
throughout  its  whole  length,  the  object  will  look  crooked  or  distorted. 
After  having  marked  in  pencil,  or  guessed  the  position  of  the  extreme 
light,  take  the  drawing  pen,  well  filled  with  a  just  perceptible  tint,  and 
draw  a  line  of  color  on  one  side  the  line  of  light,  and  almost  touching  it ; 
then  with  the  brush,  filled  with  similar  light  tint,  join  this  line  of  color 
•  whilst  still  wet,  and  fill  up  the  space  unoccupied  by  the  shade  tint,  within 
which  the  very  light  color  in  the  brush  will  disappear.  Let  that  part  of 
the  object  on  the  other  side  of  the  line  of  light  be  treated  in  the  same  way, 
and  the  desired  effect  of  a  stream  of  light  clear  and  mathematically  regu- 
lar will  be  obtained.  The  effectiveness  and  expedition  of  this  method  will 
be  most  obvious  in  coloring  long  circular  rods  of  small  diameter,  where 
the  want  of  accuracy  is  more  immediately  perceptible.  The  extreme 
depth  of  shade,  as  well  as  the  line  of  light  in  such  rods  may,  with  great 
effect,  be  indicated  by  filling  the  pen  with  dark  shade  tint,  and  drawing  it 


SHADING  AND  SHADOWS.  347 

exactly  over  the  line  representing  the  deepest  part  of  the  shade.  On 
either  side  and  joining  this  strip  of  dark  color,  another,  composed  of 
lighter  tint,  is  to  be  drawn.  Others  successively  lighter  are  to  follow, 
until,  on  one  side,  the  line  of  the  rod  is  joined,  and  on  the  other  the 
lightest  part  of  the  rod  is  nearly  reached.  The  line  of  light  is  then  to  be 
shown,  and  the  faint  tint  used  on  this  occasion  spread  with  the  brush 
lightly  over  the  whole  of  that  part  of  the  rod  situated  on  either  side  of 
this  line,  thus  blending  into  smooth  rotundity  the  graduated  strips  of  tint 
drawn  by  the  pen. 

In  all  tinted  drawings  the  more  important  parts,  whether  the  machinery 
or  the  structure,  should  be  more  conspicuously  expressed  than  those  parts 
which  are  mere  adjuncts.  Thus,  if  the  drawing  be  to  explain  the  construc- 
tion of  the  machine,  the  tint  of  edifice  and  foundations  may  be  kept  lighter 
and  more  subdued  than  those  of  tne  machine ;  and  if  the  machine,  on  the 
contrary,  be  unimportant,  it  may  be  represented  quite  light,  or  in  mere 
outline,  whilst  the  edifice  is  brought  out  conspicuously. 

As  has  been  stated,  there  are  two  methods  of  shading,  by  flat  tints  and 
by  softened  tints;  but  in  the  work,  the  "Engineer  and  Machinist's  Draw- 
ing Book,"  from  which  the  preceding  Treatise  on  Shading  and  Shadows 
has  been  taken,  the  process  of  coloring  by  flattened  tints,  or  superposition 
of  tints,  is  ignored,  and  the  method  confined  to  that  of  softened  tints ;  and 
very  strong  objection  is  made  to  washing,  although  "  a  little  washing  of  the 
most  distant  parts  is  allowable  "  (page  345).  By  this  process  recommended 
for  coloring,  a  distinct  and  even  an  artistic  drawing  of  architectural  or 
mechanical  objects  could  undoubtedly  be  made  by  a  skilful  draughtsman ; 
but  by  the  inexpert,  we  think  that  the  process  of  coloring  by  flat  tints  will 
be  found  much  the  more  simple  and  readier  way  of  producing  a  respect- 
able drawing ;  and  the  method  given  pages  330  and  331  applies  equally 
well  to  drawings  in  color. 

"With  regard  to  washings,  the  soft  sponge  is  an  implement  not  to  be 
neglected  by  the  draughtsman ;  it  is  an  excellent  means  of  correcting 
great  errors  in  drawing,  better  than  rubber  or  an  eraser,  but  care  of  course 
must  be  taken  to  wash  and  not  to  rub  off  the  surface,  and  for  errors  in 
coloring  washing  is  almost  the  only  corrector.  In  removing  or  softening 
color  on  large  surfaces,  the  sponge  is  to  be  used,  and  for  small  spots  the 
brush.  Whilst  coloring,  keep  a  clean,  moist  brush  by  you :  it  will  be  ex- 
tremely useful  in  removing  or  modifying  a  color. 

The  immediate  eifect  of  washing  is  to  soften  a  drawing,  an  effect  often 
very  desirable  in  architectural  and  mechanical  drawings,  and  the  process  is 
simple  and  easily  acquired ;  keep  the  sponge  or  brush  and  water  used 


348  SHADING  AND  SHADOWS. 

clean ;  after  the  washing  is  complete  take  up  the  excess  of  moisture  by  the 
sponge  or  brush,  or  by  a  piece  of  clean  blotting  paper.  Where  great 
vigor  is  required,  let  the  borders  of  the  different  tints  be  distinct ;  if  the 
strips  are  narrow,  the  effect  in  comparison  with  that  obtained  by  softened 
tints  is  as  a  line  engraving  compared  with  a  mezzotint. 

With  regard  to  the  colors  to  be  used  to  represent  different  materials, 
there  are  no  conventional  tints,  none  that  draughtsmen  have  agreed  upon 
to  be  uniformly  used,  and  we  think  that  some  improvement  can  be  made 
on  those  before  recommended.  India  ink,  it  has  been  observed,  is  not  a 
black,  but  a  brown,  making  with  a  blue  a  greenish  cast,  and  with  gamboge 
a  smear.  A  colored  drawing  is  better  without  the  use  of  India  ink  at  all ; 
any  depth  of  color  may  be  as  well  obtained  with  blue  as  with  black ;  there 
is  also  an  objection  to  gamboge,  that  it  is  gummy,  and  does  not  wash  well, 
and  the  effect  is  better  obtained  with  yellow  ochre.  For  the  reds,  the  mad- 
der colors  are  the  best,  as  they  stand  washing. 

For  the  shade  tint  of  almost  every  substance  a  neutral  tint,  Payne's 
grey,  or  madder  brown  subdued  with  indigo ;  for  the  local  color,  or  what 
has  improperly  been  designated  as  the  body  color,  for  cast-iron  use  a  wash 
of  indigo,  and  for  wrought-iron,  of  cobalt  blue ;  for  sections  of  these  sub- 
stances, pure  indigo  for  cast-iron,  and  cobalt  blue  for  wrought-iron,  with 
hatchings  of  deep  tints  of  the  same  color.  For  shadows  on  brass  or  cop- 
per use  madder  brown ;  local  color  for  brass,  yellow  ochre  or  Indian  yel- 
low ;  and  for  copper,  a  light  wash  of  Venetian  or  light  red ;  for  sections, 
pure  tints  or  washes,  with  deep  hatchings  of  the  same. 

For  building  material,  as  granite  or  brick,  imitate  the  color,  but  float 
on  the  tints,  leaving  it  in  patches,  and  soften  by  washes  of  clean  water,  or 
by  some  local  tints  which  suit  the  material.  The  outer  walls  of  houses,  in 
section,  are  often  colored  in  a  simple  tint  of  carmine  or  madder  brown. 
For  wood  of  a  light  color  use  a  tint  of  burnt  sienna ;  for  dark  woods,  a 
mixture  of  burnt  umber  and  sepia ;  and  for  the  shadows  madder  brown. 

Plates  LXXXVHI.  and  LXXXIX.  are  illustrations  in  chromo-lithogra- 
phy  from  colored  drawings.  It  has  not  been  possible  to  express  the  effect 
given  by  hand,  but  they  may  serve  in  a  measure  as  models,  with  the  text 
as  a  guide.  Every  one  wishing  to  become  a  draughtsman  should  make  a 
scrap-book  or  collection  of  such  drawings  as  he  can  from  time  to  time  pick 
up,  to  serve  him  as  guides,  study  the  effects  which  are  given  in  water-color 
drawings ;  in  the  architectural  department  especially  we  know  of  nothing 
cheaper  and  better  than  the  illustrations  of  some  of  the  London  papers ; 
whether  in  ink  or  color,  they  afford  capital  studies  of  design  and  of  execu- 
tion. 


TOPOGRAPHICAL   DRAWING.  34:9 


TOPOGKAPHICAL  DKAWING. 

TOPOGRAPHICAL  DRAWING  is  the  delineation  of  the  surface  of  a  locality, 
with  the  natural  and  artificial  objects,  as  houses,  roads,  rivers,  hills,  etc.,  upon 
it  in  their  relative  dimensions  and  positions ;  giving,  as  it  were,  a  miniature 
copy  of  the  farm,  field,  district,  etc.,  as  it  would  be  seen  by  the  eye  moving 
over  it.  Many  of  the  objects  thus  to  be  represented  can  be  defined  by 
regular  and  mathematical  lines,  but  many  other  objects,  from  their  irregu- 
larity of  outline,  it  would  be  very  difficult  thus  to  distinguish ;  nor  are 
the  particular  irregularities  necessary  for  the  expression.  Certain  con- 
ventional signs  have  therefore  been  adopted  in  general  use  among  drafts- 
men, some  of  which  resemble,  in  some  degree,  the  objects  for  which  they 
stand,  whilst  others  are  purely  conventional.  These  signs  may  be  ex- 
pressed by  lines,  or  by  tints,  or  by  both.  "We  commence  with  those  in 
lines,  and  in  the  latter  part  of  our  treatise,  finish  with  examples  in  color. 

Plate  XC.  fig.  I,  represents  meadow  or  grass  line,  the  J^r^-^*— rr*. 
short  lines  being  supposed  to  represent  tufts  of  grass ;  •fS'^^st^^ 
the  base  line  of  these  tufts  should  always  be  parallel  tb<5«?i*w£-S--?p 
the  base  of  the  drawing,  no  matter  what  may  be  &e^!i>C'W^*^ 
shape  of  the  enclosure.  Fig.  1  expresses  the  same  thing  M^.^J*^-*^-**- 
on  a  larger  and  coarser  scale. 

Fig.  II.  represents  an  orchard ;  fig.  III.  a  forest  or  clump  of  forest  trees. 
In  both  these  examples,  the  trees  are  represented  in  elevation ;  this  is  a 
very  common  method  of  representation,  but  not  consonant  with  the  other 
parts  of  the  plan.  It  seems  better  that  trees  should  be  represented  in 
plan,  as  in  fig.  TV.  Orchards  may  be  represented  thus  (fig.  2),  and  forests, 
on  a  larger  scale,  by  a  sort  of  distinctive  foliage,  according  to  the  kinds  of 
trees ;  thus  fig.  3  may  represent  chestnut,  fig.  4  oak,  fig.  5  pine  and  fir. 
When  trees  occur  upon  a  hill-side,  the  shading  lines  of  the  hill-side  should 
be  interrupted  to  receive  the  body  of  the  tree,  but  not  its  shadow,  which 
may  be  drawn  independently  of  them  when  the  slope  is  slight,  but  when 


350 


TOPOGKAPHICAL   DRAWING. 


it  is  steep  the  shadows  may  be  omitted,  and  the  trees  shaded  nearly  as 
dark  as  that  of  the  slope,  but  the  foliage  should  be  represented  rather 
sparse. 


Fig.  2. 


Fig.  3. 


Fig.  4. 


Fig.  5. 


Fig.  V.  represents  a  house  and  cultivated  ground ;  the  walks  and  roads 
are  in  white,  the  buildings  are  marked  by  diagonal  lines.  The  cultivated 
land  by  parallel  rows  of  broken  and  dotted  lines,  supposed  to  be  furrows. 
Sometimes  signs  are  used  to  represent  the  crops. 

Fig.  VI.  represents  marsh  land,  water  and  bog.  Fig.  VII.,  a  river 
with  mud  and  sand  banks.  Sand  is  represented  by  fine  dots  made  with 
the  point  of  the  pen ;  mud  in  a  very  similar  way,  but  the  dots  should  be 
much  closer  together..  Gravel  is  represented  by  still  coarser  dots,  and 
stones  by  irregular  angular  forms,  imitating  their  appearance,  as  seen  from 
above. 

Fig.  VEIL  represents  a  bold  shore  bounded  by  cliffs.  "Water  is  almost 
invariably  represented  in  the  same  way,  except  in  connection  with  bogs 
(fig.  VI.),  by  drawing  a  line  parallel  to  the  shore  or  coast,  following  its 
windings  and  indentations,  and  as  close  to  it  as  possible;  then  another 
parallel  a  little  more  distant,  a  third  still  more  so,  and  so  on.  Small  ponds 
are  sometimes  represented  by  parallel  horizontal  lines,  but  usually  by  the 
curved  lines  of  shore.  Brooks,  and  even  rivers,  when  the  scale  is  small, 
are  represented  by  one  or  two  lines.  The  direction  of  the  current  is  shown 
by  arrows. 

Fig.  6  represents  a  turnpike.  If  the  toll-bar  and  marks  for  a  gate 
be  omitted,  it  is  a  common  highway.  Fig.  7  represents  a  road  as  sunk  or 
cut  through  a  hill.  Fig.  8,  one  raised  upon  an  embankment.  Fig.  9  is  a 


Fig.  6. 


Fig.  7. 


Fig.  8. 


Fig.  9. 


railroad,  often  represented  without  the  cross-tie,  by  two  heavy  parallel 
lines,  sometimes  by  but  one. 


TOPOGRAPHICAL    DRAWING.  351 

Fig.  10  represents  a  bridge  with  a  single  pier.     Fig.  11,  a  swing  or 


JL 


Fig.  10. 


Fig.  11. 


Fig.  12. 


Fig.  13. 


Fig.  14 


draw  bridge.  Fig.  12,  a  suspension  bridge,  and 
fig.  13  a  ford.  Fig.  14,  a  lock  of  a  canal.  Ca- 
nals are  represented  like  roads,  except  that  in 
the  latter  the  side  from  the  light  is  the  shaded 
line,  in  the  former,  the  side  to  the  light. 

Fig.  15   represents  dwellings,  or  edifices  of         Fig.  is.  Fig.  ie. 

any  sort ;  they  are  often  made  distinctive  of  their  purpose  by  some  small 
prefix,  as  a  pair  of  scales  for  a  court-house,  an  elevation  of  a  sign-post  for 
a  tavern,  a  letter  for  a  post-office,  a  horseshoe  for  a  smithy,  a  small  water- 
wheel  for  a  water-mill,  and  a  chimney  for  a  steam-mill. 

Fig.  16  represents  a  church  or  cathedral ;  this  is  sufficiently  expressed 
by  its  plan ;  but  usually,  churches  are  represented  according  to  their  own 
plan,  with  the  distinctive  prefix  of  a  cross  or  a  steeple. 

The  localities  of  mines  may  be  represented  by  the  signs  of  the  planets, 
which  were  anciently  associated  with  the  various  metals,  and  a  black  circle 
for  coal.  Thus  £  Mercury,  ?  Copper,  T?  Lead,  D  Silver,  O  Gold,  $  Iron, 
It  Tin,  •  Coal. 

On  the  Representation  of  Hills. — The  two  methods  in  general  use  for 
representing  with  a  pen  or  pencil  the  slopes  of  ground,  are  known  as  the 


Fig.  IT 


Fig.  13. 


352 


TOPOGRAPHICAL   DRAWING. 


vertical  and  the  horizontal.  In  the  first  (fig.  17),  the  strokes  of  the  pen 
follow  the  course  that  water  would  take  in  running  down  these  slopes. 
In  the  second  (fig.  18),  they  represent  horizontal  lines  traced  round  them, 
such  as  would  be  shown  on  the  ground  by  water  rising  progressively  by 
stages,  1,  2,  3,  4,  5,  6,  up  the  hill.  The  last  is  the  most  correct  represen- 
tation of  the  general  character  and  features  of  the  ground,  and  when  ver- 
tical levels  or  contours  have  been  traced  by  level  at  equal  vertical  dis- 
tances over  the  surface  of  the  ground,  they  should  be  so  represented ;  or 
when,  by  any  lines  of  levels,  these  contours  can  be  traced  on  the  plans 
with  accuracy,  the  horizontal  system  should  be  adopted  ;  but  where,  as  in 
most  plans,  the  hills  are  but  sketched  in  by  the  eye,  the  vertical  system 
should  be  adopted,  it  affords  but  proximate  data  to  judge  of  the  slope, 
whereas,  by  the  contour  system,  the  slope  may  be  measured  exactly.  It  is 
a  good  maxim  in  topographical  drawing,  not  to  represent  as  accurate  any 
thing  which  has  not  been  rigorously  established  by  surveys.  On  this 
account,  for  general  plans,  when  the  surface  of  the  ground  has  not  been 
levelled,  nor  is  required  to  be  determined  Avith  mathematical  precision, 
we  prefer  the  vertical  to  the  horizontal  system  of  representing  slopes. 

On  drawing  hills  on  the  vertical  system,  it  is  very  common  to  draw 
contour  lines  in  pencil  as  guides  for  the  vertical  strokes.  If  the  horizontal 
lines  be  traced  at  fixed  vertical  intervals,  and  vertical  strokes  be  drawn 
between  them  in  the  line  of  quickest  descent,  they  supply  a  sufficiently  ac- 
curate representation  of  the  face  of  the  country  for  ordinary  purposes.  It 
is  usual  to  make  the  vertical  strokes  heavier  the  steeper  the  inclination, 
and  systems  have  been  proposed  and  used,  by  which  the  inclination  is 
defined  by  the  comparative  thickness  of  the  line  and  the  intervening 
spaces. 

In  describing  ground  with  the  pen,  the  light  is  generally  supposed  to 
descend  in  vertical  rays,  and  the  illumination  received  by  each  slope  is  di- 
minished in  proportion  to  its  divergence  from  the  plane  of  the  horizon. 

Thus  in  fig.  19,  it  will  be  seen 
that  a  horizontal  surface  receives 
an  equal  portion  of  light  with  the 
inclined  surface  testing  upon  it, 
and  as  the  inclined  surface  is  of 

Fls  19.  greater  extent,  it  will  be  darker 

than  the  horizontal  in  proportion  to  the  inclination  and  consequent  in- 
crease of  the  surface,  and  on  this  principle  varied  forms  of  ground  arc 
represented  by  proportioning  the  thickness  of  stroke  to  the  steepness  of 
the  slope. 


TOPOGRAPHICAL   DRAWING. 


353 


In  the  German  system  as  proposed  by  Major  Lehmann,  of  representing 
the  slopes  of  ground  by  a  scale  of  shade,  the  slope  at  an  angle  of  45°,  as 
reflecting  its  light  horizontally,  is  supposed  to  be  the  greatest  ever  required 
to  be  shown,  and  is  represented  by  black,  whilst  the  horizontal  plane  re- 
flecting all  rays  upward  is  represented  by  white,  'and  the  intermediate 
slopes  by  different  proportions  of  black  in  the  lines  to  white  in  the  spaces 
intervening.  "  We  have  not  thought  it  necessary  to  give  an  illustration  of 
this  scale  of  shade,  as  it  does  not  discriminate  between  slopes  of  greater 
inclination  than  45°,  preferring  the  modification  as  proposed  for  the  IT.  S. 
Coast  Survey,  adapted  to  the  representation  of  all  necessary  slopes,  and 
consonant  with  the  demonstration,  fig  19.  Fig.  20  represents  this  scale  of 


shade  tabellated,  the  following  are  the  proportions  of  black  and  white  for 
different  inclinations,  and  the  construction  may  be 
easily  understood  from  fig  20.  Thus  form  eight  paral- 
lel rectangles  according  to  the  number  of  slopes  to 
be  represented ;  divide  each  of  these  rectangles  into 
eleven  parts,  then  the  proportion  of  white  to  black  in  a 
slope  of  2i°  will  be  to  make  one  of  these  parts  black  ;• 
of  5°  two  parts,  of  10°  three,  and  so  on.  Now  thicken 
the  lines  according  to  this  proportion,  and  copy  the 
strokes  till  the  hand  becomes  habituated  to  their  for- 
mation, and  the  eye  so  practised,  that  the  graduation  for  all  practical  pur- 
poses may  be  performed  without  direct  reference  to  the  scale. 

On  Drawing  Hills  ly  Contour s.^-Draw  first  the  curves  which  have 
been  traced  on  the  ground  by  levels,  and  these  should  be  distinguished 
from  the  other  lines  by  color,  as  red,  or  by  size  of  lines.  It  should  be  ob- 
served that  whatever  point  has  been  actually  established  by  survey,  it 
should  not  be  confounded  with  sketching  by  eye.  If  there  are  no  such 
lines,  but  it  is  merely  intended  to  sketch  the  hills  as  in  the  usual  vertical 
style,  lay  off  the  curves  at  equal  vertical  intervals,  say  10,  20,  50  or  100 
feet,  according  to  the  scale,  and  then  proceed  to  fill  in.  The  ground  between 
23 


354: 


TOPOGRAPHICAL   DRAWING. 


these  fixed  curves  or  sections,  is  supposed  to  slope  uniformly.  Divide  the 
space  therefore  equally,  and  draw  within  each  set  of  curves  as  many  lines 
as  may  be  suited  to  the  scale  of  the  map,  and  the  vertical  intervals  be- 
tween the  curves.  Draw  the  lines  with  firmness,  and  let  them  have  a 
length  varying  from  one  to  three  fourths  of  an  inch,  according  to  the  great- 
er or  less  degree  of  the  slope.  When  the  hill  is  steep  the  frjies  should  be 
short  and  heavy,  growing  longer  and  lighter  as  the  inclination  becomes 
less.  The  lines  should  nearly  touch  each  other,  so  as  to  appear  almost 
consecutive,  but  not  overlap,  nor  with  a  determinate  interval  between 
their  ends.  Fig.  21  represents  the  half  of  the  hill,  fig.  18,  and  at  double 
scale,  completed  by  drawing  the  intermediate  contour  lines. 


Fig.  21. 


Drawing  hills  by  contours  is  of  comparatively  late  introduction,  and 
is  generally  practised  abroad,  but  little  used  here ;  it  is  more  difficult  for 
the  draughtsman,  and  no  more  expressive  of  the  features  of  the  ground 
than  the  vertical  system,  and  has  little  to  recommend  except  where  actual 
lines  have  jeen  traced,  and  it  becomes  a  record  of  facts.  Certain  lines 
in  pencil  are  necessary  for  the  ^i'oper  drawing  according  to  the  vertical 
system,  but  when  the  drawing  is  complete,  an  implied  line  is  merely  left. 
Hills  are  much  more  effectively  expressed  by  the  brush  than  the  pen,  and 
much  more  readily,  of  which  illustrations  will  be  given  further  on. 

In  our  list  of  conventional  signs  we  have  given  but  few,  and  I 
only  the  most  prominent.  It  is  useless  to  tax  the  memory  with  many,  as 
the  purposes  for  which  an  edifice  or  locality  is  intended  will  supply  some 
characteristic  by  which  they  are  easily  distinguished ;  as  in  case  of  mills 
already  given,  or  as  in  case  of  a  graveyard  by  a  tombstone,  a  quarry  by 
a  stone-hammer,  a  battle  field  by  crossed  swords,  &c.  "When  there  is  no 
obvious  characteristic,  the  positions  may  be  lettered  or  numbeix.l  and  ex- 
plained bv  marginal  notes,  if  there  be  not  room  on  the  plan  in  its  appro- 
priate locality. 


TOPOGRAPHICAL   DEAWING.  355 


PLOTTING. 

Plotting  is  the  making  of  the  plan  on  paper  from  the  measurements 
taken  in  the  field. 

The  rough  sketch  is  usually  made  in  the  field-book,  that  is,  the  book 
kept  in  the  field,  in  which  all  the  steps  or  observations  of  the  survey  are 
noted  on  the  spot.  The  field-book  is  generally  ruled  with  a  middle  col- 
umn, from  one  half  to  one  inch  in  width.  This  middle  column  is  intended 
to  represent  the  station  line  itself,  and  all  lines  crossing  the  station  line 
are  not  drawn  directly  across  the  middle  column,  but  arrive  at  one  side 
and  leave  it  on  the  other,  at  points  precisely  opposite.  The  middle  column 
is  reserved  entirely  for  the  angles  and  measures,  made  in  direction  of  the 
station  line.  All  measurements  of  offsets  or  angles  other  than  those  on 
the  direct  line  are  entered  in  the  marginal  spaces  at  each  side  of  the  middle 
column,  according  to  the  side  of  the  station  line  on  which  they  are  taken. 
The  stations  are  marked  thus  0/and  the  notes  commence  at  the  bottom  of 
the  page. 

Scales. — The  choice  of  the  scale  for  the  plot  depends  in  a  great  measure 
on  the  purpose  for  which  the  plan  is  intended.  It  should  be  large 
enough  to  express  all  the  details  which  it  is  desirable,  modified  by  the  cir- 
cumstances, whether  the  map  is  to  be  portable,  or  whether  space  can  be 
afforded  for  the  exhibition  of  a  l^rge  plan.  We  must  adapt  our  plan  for 
the  purposes  which  it  is  intended  to  illustrate,  and  the  place  it  is  to  oc- 
cupy. 

Plans  of  house  lots  are  usually  named  as  being  so  many  feet  to  the 
inch ;  plots  of  farm  surveys,  as  so  many  chains  to  the  inch ;  maps  of  sur- 
veys of  States,  as  so  many  miles  to  the  inch,  and  maps  of  railway  surveys, 
as  so  many  feet  to  the  inch,  or  so  many  inches  to  the  mile. 

For  farm  surveys,  if  of  small  extent,  two  chains  to  the  inch  is  a  con- 
•nt  scale ;  for  larger  farms  three  chains  to  the  inch.  This  last  seal"  is 
that  prescribed  by  the  English  Tithe  Commissioners  for  the  first-cL....; 
maps.  One  acre  laid  out  in  the  form  of  a  square,  to  the  scale  of 

1  chain  to  the  inch  occupies       3.16  inches  square. 
U     "          "  "  2.10     "          « 

2  «          «  «  1.58     "          « 

3  "  "  "  1.05     "  " 

and  so  on. 


356  TOPOGRAPHICAL   DRAWING. 

Knowing  how  much  is  the  area  of  the  ground  to  be  plotted,  if  the  form 
is  square  we  can  easily  determine  the  side  of  the  square  occupied,  by  mul- 
tiplying the  square  root  of  the  area  in  acres  by  3.16,  and  dividing  the 
product  by  the  number  of  chains  to  the  inch  in  the  scale  assumed.  Thus 
if  50  acres  were  to  be  plotted  in  a  square,  to  the  scale  of  3  chains  to  the 

OO  Q_l 

inch  =  V50  =  7.07.     7.07  x  3.16  =  22.34      -~-  —  7.45   inches,   side 

o 

of  the  square  of  the  plot  on  a  scale  of  3  chains  to  the  inch.  This  rule  will 
assist  the  draughtsman  in  selecting  a  scale  for  figures  not  very  irregular  in 
form. 

State  surveys  are  of  course  plotted  on  a  smaller  scale  than  those  of 
farms.  On  the  U.  S.  Coast  survey  all  the  scales  are  expressed  fractionally 
and  decimally.  The  original  surveys  are  generally  on  a  scale  of  one  to  ten 
or  twenty  thousand,  but  in  some  instances  the  scale  is  larger  or  smaller. 
The  public  surveys  embrace  three  general  classes: — 1.  Small  harbor 
charts.  2.  Charts  of  bays,  sounds,  &c.  3.  General  coast  charts. 

The  scales  of  the  first  class  vary  from  1 :  5,000  to  1 : 60,000,  according 
to  the  nature  of  the  harbor  and  the  different  objects  to  be  represented. 

The  scale  of  the  second  class  is  usually  fixed  at  1 :  80,000.  Prelimi- 
nary charts  are,  however,  issued  of  various  scales,  from  1 : 80,000  to 
1 :  200,000. 

Of  the  third  class  the  scale  is  fixed  at  1 :  400,000  for  the  general  chart 
of  the  coast  from  Gay  Head  to  Cape  Henlopen,  although  considerations 
of  the  proximity  and  importance  of  points  on  the  coast  may  change  the 
scales  of  charts  of  other  portions  of  our  extended  coast. 

On  all  plots  of  large  surveys,  it  is  very  desirable  that  the  scales  adopted 
should  bear  a  definite  numerical  proportion  to  the  linear  measurement  of 
the  ground  to  be  mapped,  and  that  this  proportion  should  be  expressed 
fractionally  on  the  plan,  even  if  the  scale  be  drawn  or  expressed  some 
other  way,  as  chains  to  the  inch.  The  decimal  system  has  the  most  to 
recommend  it,  and  is  generally  adopted  in  government  surveys. 

For  Eailroad  Surveys,  the  New  York  general  railroad  law  directs  the* 
scale  of  map  which  is  to  be  filed  in  the  State  engineer's  office,  to  be  500 
feet  to  one-tenth  of  a  foot,  1 :  5,000. 

For  the  Canal  Maps,  a  scale  of  2  chains  to  the  inch,  1 : 1584  is  em- 
ployed. In  England,  plans  and  sections  for  projected  lines  of  inland  com- 
munication, or  generally  for  public  works  requiring  the  sanction  of  the 
Legislature,  are  required,  by  the  "  standing  orders,"  to  be  drawn  to  scales 
not  less  than  4  inches  to  the  mile,  1 : 15,840,  for  the  plan,  and  100  feet  to 
the  inch,  1 : 1,200,  for  the  profiles. 


TOPOGRAPHICAL   DRAWING.  357 

In  the  United  States  engineer  service,  the  following  scales  are  pre- 
scribed : 

General  plans  of  buildings, 10  feet  to  the  inch,  1  :  120 

Maps  of  ground  with  horizontal  curves  1  foot  apart,  50      "  "        1  :  600 

Topographical  maps  1£  miles  square,      .         .         .1  mile  to  2  feet,  1  :  2,610 

"      comprising  3  miles  square,      .1        "  1  foot,  1  :  5,280 

"  "  "  bet.  4  and  8  miles,    1        "  6  in.,    1  :  10,560 

"  "  9  miles  square,      .1        "  4    "      1  :  15,840 

Maps  not  exceeding  24  miles  square,       .         .         .1        "  2    "      1  :  31,680 

"      comprising      50     "         "  .         .     '    .    1        "  1    "      1  :  63,360 

100     "         "  ...         .1        "  i    "      1  :  126,720 

Surveys  of  roads  and  canals,  .         .         .         .50  feet  to  1    "      1  :  600 

The  description  of  various  scales  and  the  principles  of  their  construc- 
tion will  be  found  at  pp.  17  and  18,  to  which  the  reader  is  referred. 

In  plotting  from  the  field  book,  the  first  lines  to  be  laid  down  are  the 
outlines  or  main  lines  of  the  survey.  If  the  survey  has  been  made  by 
triangles,  the  principal  triangles  are  first  laid  down  in  pencil  by  the  inter- 
section of  their  sides,  the  length  being  taken  from  the  scale  and  described 
with  compasses ;  if  the  lines  are  longer  than  the  reach  of  the  compasses, 
or  the  extent  of  the  scale,  lay  off  the  length  on  any  convenient  line,  and 
measure  and  describe  with  beam  compasses.  The  principal  triangles  being 
laid  down,  other  points  will  be  determined  by  intersections  in  the  same 
manner  as  measured  on  the  ground.  In  general,  when  the  surveys  have 
been  conducted  without  instruments  to  measure  the  angles,  as  the  compass 
or  theodolite,  the  position  of  the  points  on  paper  are  determined  by  the 
intersection  and  construction  of  the  same  lines  as  has  been  done  in  the 
field. 

Surveys  are  mostly  conducted  by  measuring  the  inclination  of  lines  to 
a  meridian  or  to  each  other  by  the  compass,  or  by  the  theodolim  In  the  sur- 
veys of  farms,  where  great  accuracy  is  not  required,  the  compass  is  most 
used.  The  compass  gives  the  direction  of  a  line  in  reference  to  the  magne- 
tic meridian.  The  variation  from  the  true  meridian,  or  a  direct  north  and 
south  line,  varies  considerably  in  different  parts  of  the  country.  In  1840, 
the  line  of  variation  in  which  the  needle  pointed  directly  north,  passed 
in  a  nearly  straight  direction  from  a  little  west  of  Cape  Hatteras, 
N.  C.,  through  the  middle  of  Virginia,  about  midway  between  Cleve- 
land, Ohio,  and  Erie,  Pa.,  and  through  the  middle  of  lakes  Erie  and 
Huron.  At  all  places  east  of  this  line  the  variation  is  westerly,  that  is,  the 
needle  points  west  of  the  line  north.  West  of  this  line  the  variation  is 
'easterly. 


358 


TOPOGRAPHICAL   DRAWING. 


Fig.  22  represents  the  field  notes  of  a  survey  by  compass.  Fig.  23  a 
plot  of  the  same,  with  the  position  of  the  protractor  in  laying 
off  the  angles.  In  this  way  of  plotting,  a  meridian  is  laid  off 
at  the  intersection  of  each  set  of  lines.  Sometimes  the  angles 
are  plotted  directly  from  the  determination  of  the  angle  of  de- 
flection of  two  courses  meeting  at  any  point,  without  laying 
down  more  than  one  meridian :  Figure  24.  When  the  first 


3.23 


-(5)- 
3.54 


cn 

-(4)- 
2.22 


-(3)- 
1.29 


-(2)- 
2.78 


Fig.  23. 


letters  of  the  bearing  are  alike,  that  is,  both  K.  or  both  S.,  and  the  last  let- 
ters also  alike,  both  E.  or  both  W.,  the  angle  of  deflection  C  B  B'  will  be 
the  difference  of  the  bearings,  or,  in  this  instance,  20°. 


w- 


N 


Fig.  24. 


'  Fig.  25.     When  the  first  letters  are  alike  and  the  last  different,  the 
angle  C  B  B'  will  be  the  sum  of  the  two  bearings. 


TOPOGRAPHICAL   DRAWING. 


Eig.  26.  When  the  first  letters  are  different  and  the  last  alike,  sub- 
tract the  sum  of  the  bearings  from  180°  for  the  angle  C  B  B' :  when  both 
the  first  letters  and  last  are  different,  subtract  their  difference  from  180° 
for  the  angle. 

Instead  of  drawing  a  meridian  through  each  station,  or  laying  off  the 
angle  of  deflection,  by  far  th^easiest  way  is  to  lay  off  but  a  single  meridian 
near  the  middle  of  the  sheet ;  lay  off  all  the  bearings  of  the  survey  from 
some  one  point  of  it  as  shown  in  fig.  27,  and  number  to  correspond 
with  the  stations  from  which 
the  bearings  are  taken,  and 
then  transfer  them  to  the 
places  where  they  are  wanted 
by  any  of  the  instruments  used 
for  drawing  parallel  lines.  For 
the  protracting  of  the  rough 
plan,  sheets  of  drawing  paper 
can  be  bought  with  protractors 
printed  on  them.  When  the 
plans  are  large,  it  is  often 
convenient  to  lay  out  two  or 
three  meridians  on  different 
parts  of  the  sheet  and  lay  off 
the  bearings  of  lines  adjacent  to  each  meridian  upon  them. 

In  plotting  from  a  survey  by  a  theodolite  or  transit,  it  is  generally 
usual  to  lay  off  the  angles  of  deflection  of  the  different  lines  as  taken 
in  the  field,  plotting  all  the  tie  lines  as  corrections. 

When  the  plot  of  a  survey  does  not  close,  that  is,  come  together,  or 
return  to  the  point  of  commencement,  as  it  seldom  does  exactly,  it  may 
be  corrected  o.r  forced ;  but  first  be  sure  that  the  bearings  and  distances  as 
given  in  the  field  book  are  laid  down  accurately,  and  then  proceed  to  cor- 
rect as  follows :  thus,  let  A  B  0  D  E, 
fig  28,  be  the  boundary  lines  plotted 
according  to  the  notes,  and  suppose 
the  last  course  conies  to  E  instead 
of  ending  at  A  as  it  should.  Sup- 
pose also  that  there  is  no  reason  to 
suspect  any  error  more  in  one  line 
than  another,  that  the  measures 
and  bearings  of  all  are  equally  cer-  FJ-.  2^- 


»  360 


TOPOGRAPHICAL   DRAWING. 


tain;  then  the  inaccuracy  must  be  distributed  among  all  the  lines  in 
proportion  to  their  length.  Each  point,  B,  C,  D,  E,  must  be  moved  in  a 
direction  parallel  to  E  A,  by  a  certain  distance.  Thus  add  together  the 
length  of  all  the  lines,  and  this  sum  is  to  the  line  A  B,  as  the  error  A  E  is 
to  the  correction  B  B' ;  for  the  next  point,  the  whole  sum  is  to  A  B,  B  C,  as 
the  error  is  to  the  correction,  E  C ;  and  so  o^ ;  obtaining  the  second  term 
of  the  proportion  by  adding  consecutively  the  different  lines.  This  calcu- 
lation may  be  much  simplified  by  the  use  of  the  sector,  according  to  the 
rule  given  for  finding  a  fourth  proportional  (p.  23).  Take  the  error  A  E 
from  the  plan,  and  open  the  sector  until  this  quantity  becomes  the  trans- 
verse distance  of  the  first  term  or  sum  of  the  lines;  then  the  distance  be- 
tween the  points  corresponding  to  the  consecutive  sums  will  be  the  corre- 
sponding error. 

The  best  way  of  correcting  errors  and  of  plotting  a  survey,  whether 
made  by  compass  or  by  theodolite,  is  by  balancing  the  survey,  or  correct- 
ing the  latitudes  and  departure  of  the  courses  so  that  they  shall  be  equal. 
For  the  method  of  doing  this,  we  refer  to  any  of  the  recent  works  on  sur- 
veying. From  Gillespie  on  Land  Surveying,  we  have  taken  most  of  the 
preceding  on  the  plotting  of  angular  surveys,  and  the  following  para- 
graph on  balancing. 


Ita. 

Total  Latitude 
from  Sta.  1. 

Total  Departure 
from  Sta.  1. 

1 

0.00 

0.00 

2 

+  2.21  N. 

+  1.55E. 

f 

+  2.36  N. 

+  2.S3E. 

4 

+  1.15N. 

+  4.69  E. 

5 

-  1.78  S. 

+  2.69  E. 

1 

0.00 

0.00 

This  table  represents  the 
survey  as  given,  fig.  22,  bal- 
anced. 

To  plot  from  this  table, 
draw  a  meridian  through  the 
point  taken  for  station  1,  as 
in  fig.  29.  Set  off  upward 
from  this  along  the  meridian 
the  latitude  2.21  chains  north 
to  A,  and  from  A  to  the  right  or  E.  set  off  the  departure  1.55  chains.  This 
gives  the  point  or  position  of  Station  2,  join  1  and  2.  From  A  again  set  off 
upward  2.36  chains,  and  from  B  to  the  right  perpendicularly  set  off 
2.83  chains,  which  gives  position  of  Station  3,  join  2  and  3 :  and  so  pro- 
ceed setting  off  North  latitudes  upwards,  and  South  downwards,  East 


TOPOGRAPHICAL   DRAWING. 


361 


departures  perpendicularly  to  the  right, "and  West  perpendicularly  to  the 
left. 

In  balancing  surveys  made  by  a  theodolite,  a  meridian  is  assumed, 
generally  one  of  the  lines  of  the  survey.  The  most  convenient  will  be 
some  long  line  of  which  ^he  survey  lies  all  to  one  side. 


Fig  30. 

The  advantages  of  this  method  of  plotting  are 
its  accuracy,  rapidity,  the  impossibility  of  an  er- 
ror in  one  point  affecting  the  others,  and  the  cer- 
tainty of  coming  together. 

The  above  explains  the  method  of  plotting  the 
main  lines  of  the  survey ;  the  filling  in  is  from 
points  established  from  these  main  lines,  either 
by  the  construction  of  triangles,  by  measure,  or 
by  angles,  or  by  perpendiculars.  In  case  of  un- 
important lines,  as  the  crooked  brook  for  instance, 
fig.  30,  offsets  are  taken  to  the  most  prominent 
angles,  as,  a,  a,  a,  and  the  intermediate  bends  are 
sketched  by  eye  into  the  field  book.  In  copy- 
ing them  on  the  plan  a  similar  construction  is 
adopted. 

The  most  rapid  way  of  plotting  the  offsets,  is 
by  the  use  of  a  plotting  and  offset  scale,  fig.  31 — 
the  one  being  fixed  parallel  to  the  line  A  B  from 
which  the  offsets  are  to  be  laid  off,  at  such  a  dis- 
tance from  it,  that  the  zero  line  on  the  movable 
scale  coincides  with  it,  whilst  the  zero  of  its  own 
scale  is  on  a  line  perpendicular  to  the  position  of 
the  station  A  from  which  the  distances  were  meas- 
ured. It  is  to  be  observed  that  in  the  field  book 
all  the  measures  are  referred  to  the  point  of  be-  Fig.  si. 

ginning  on  any  one  straight  line.  Having  placed  the  plotting  scale,  move 
the  offset  scale  to  the  first  distance  by  the  scale  at  which  an  offset  has 
been  taken,  mark  off  now  on  the  offset  scale  the  length  of  the  offset  on 


362  TOPOGRAPHICAL   DRAWING. 

its  corresponding  side  of  the  liner.  Proceed  then  to  the  next  distance,  es- 
tablishing thus  repeated  points,  join  the  points  by  lines  as  they  are  on  the 
ground. 

The  plotting  and  offset  scale  must  of  course  be  of  the  same  scale  as  the 
rest  of  the  drawing,  on  which  account  it  may  not  always  be  possible  to 
obtain  such  scales  adapted  to  those  of  the  plan ;  but  they  may  be  easily 
constructed  of  thick  drawing  paper  or  pasteboard. 

When  a  great  deal  of  plotting  to  one  scale  is  necessary,  as  in  govern- 
ment surveys,  the  offset  scale  may  be  made  to  slide  in  a  groove  upon  the 
plotting  scale. 

In  protracting  the  triangles  of  an  extended  trigonometrical  survey  in 
which  the  sides  have  been  calculated  or  measured,  it  is  better  to  lay  down 
the  triangles  from  the  length  of  their  sides  rather  than  by  measuring  the 
angles,  because  measures  of  length  can  be  taken  with  more  accuracy  from 
a  scale,  and  transferred  to  the  plan  with  more  exactness  than  angles  can 
be  pricked  off  from  a  protractor ;  but  for  ordinary  surveys,  the  triangula- 
tion  is  most  frequently  and  expeditiously  plotted  by  the  means  of  a  pro- 
tractor. 

The  outlines  of  the  survey  having  been  balanced  and  plotted  in,  and 
the  subsidiary  points,  as  established  by  offsets  and  by  triangles,  the  filling 
in  of  the  interior  detail  is  done  by  copying  from  the  field  book  the  natural 
features  of  the  ground,  in  their  appropriate  position,  and  according  to  the 
conventional  signs  already  described. 

In  many  surveys,  as  of  roads,  rivers,  canals,  and  boundaries,  the  plot 
to  be  made  is  but  a  single  line,  with  a  few  of  the  nearest  local  objects  on 
either  side.  In  some  instances  the  angles  at  each  intersection  are  taken 
merely  with  reference  to  the  two  lines  forming  the  angle,  and  are  therefore 
to  be  plotted  as  shown  in  fig.  23,  by  laying  the  protractor  at  each  inter* 
section ;  but  in  other  instances,  by  the  method  of  surveying,  the  directions 
of  all  lines  are  referred  to  the  first  as  meridian,  or  if  the  survey  is  exten- 
sive, to  some  number  of  lines,  and  the  plotting  is  then  expeditiously  per- 
formed as  in  fig.  27.  In  this  system  of  surveying,  instead  of  fixing  the 
vernier  at  zero,  for  every  back  angle  the  preceding  forward  angle  is  re- 
tained except  for  those  lines  intended  as  meridians. 

Surveys  for  railways,  like  those  above,  are  of  lines  extensive  in  length 
but  of  very  little  width.  In  the  surveys  of  preliminary  or  trial  lines,  the 
curves  at  the  intersection  of  lines  are  seldom  introduced  ;  an(t  in  plotting 
1  it  is  but  the  usual  method  of  plotting  surveyed  lines,  by  either  of  the 
methods,  according  as  the  survey  may  have  been  conducted,  with  the  the- 
odolite, or  with  the  compass. 


TOPOGRAPHICAL   DRAWING. 


363 


In  plotting  curves  on  a  line  of  location,  lay  off  from  the  intersection 

f  tangents,  as  C,  fig.  32,  the  distance  of  the  tangent  points  A  and  B,  and 

find  the  centre  O  of  the  curve,  by  the  erection  of  perpendiculars  to  these 


Degree. 

Radii,  ft. 

Central  Ordinate. 

1° 

5T29.65 

0.218 

20 

286493 

0.436 

30 

1910.08 

0.655 

40 

1432.69 

0.873 

50 

1146.28 

1.091 

63 

955.8T 

1.309 

TO 

819.02 

152S 

80 

T16.7S 

1.74G 

90 

63T.27 

1.965 

100 

573.69 

2.183 

two  points,  or  if  the  ra- 
dius of  the  curve  is 
known,  by  describing 
arcs  with  this  radius 
from  the  same  points. 
Railway  curves  are  de- 
signated by  degrees  or 


according  to  the  angle   of  deflection  made   by  two  '  chords  of  100  feet 
each. 

Two  curves  often  succeed  each  other  having  a  common  tangent,  at  the 
point  of  junction.  If  the  curves  lie  on  opposite  sides  of  the  common  tan- 
gent, they  form  a  reversed  curve,  ABC,  (fig.  33,)  and  their  radii  may  be 

the  same  or  different. 
If  they  lie  on  the  same 
side  of  the  common 
tangent  and  have  dif- 
ferent radii,  they  form 
a  compound  curve,  A 
BD.  When  the  radii  of 
curvature  are  known, 
the  determination  of 
the  centres  is  ob- 
tained easily,  by  de- 
scribing arcs  with  the  established  radii  from  the  tangent  points. 

If  the  radii  of  curvatures  are  not  known,  and  it  is  required  to  plot 
a  compound  or  reversed  curve  which  shall  be  tangent  at  the  points  A 
and  D  or  C  to  other  straight  lines,  the  change  of  curvature  taking 


364r 


TOPOGRAPHICAL   DRAWING. 


plac*e  at  B,  then  at  A  erect  a  perpendicular  to  the  given  line,  find 
some  point  on  this  perpendicular  which  is  equally  distant  from  A  and  B, 
and  this  point  will  be  the  centre  of  the  curve  A  B ;  through  this  point 
and  B  draw  an  indefinite  line,  its  intersection  by  the  perpendicular  to  the ' 
tangent  at  D  will  be  the  centre  for  the  other  potion  of  the  compound 
curve  ;  and  its  intersection  by  the  perpendicular  to  the  tangent  at  C  will 
give  the  centre  for  the  reversed.  The  centres  of  curves  tangent  to  each 
other  must  lie  in  a  straight  line,  passing  through  their  point  of  con- 
nection. 

When  the  curves  are  larger  than  can  be  described  by  the  dividers  or 
beam  compasses,  they  can  be  plotted  as  shown  in  geometrical  problems, 
or  points  of  a  curve  may  be  obtained  by  calculation  of  their  ordinates, 
and  the  curves  drawn  from  point  to  point  by  sweeps  and  variable  curves. 
Approximately,  knowing  the  central  ordinate  of  the  curve  between  two 
points,  the  central  ordinate  of  one  half  that  curve  will  be  one  quarter  of 
the  first.  Thus,  fig  32,  G  D  is  about  one  quarter  of  E  F,  hence  by  subdi- 
visions as  many  points  as  are  necessary  may  be  obtained ;  but  it  should 
be  observed,  that  the  greater  the  number  of  degrees  in  the  arc,  the  less 
near  to  the  truth  the  rule. 

w  3l.43FcetFattpertiae  x/.  Ltvel  ^ 


Fig.  35  represents  a  plot  of  a  railway  line ;  in  this  plot  the  curve  is 
represented  as  a  straight  line,  the  radius  of  curvature  being  written  in. 
This  method  is  sometimes  adopted  when  it  is  desirable  to  confine  the  plot 
within  a  limited  space  upon  the  sheet,  and  it  is  convenient  when  plotted 
thus  directly  beneath  the  profile  or  longitudinal  section  (fig.  34). 

In  plotting  the  section,  a  horizontal  or  base  line  is  drawn  on  which  are 
laid  off  the  stations-  or  distances  at  which  levels  have  been  taken ;  at  these 
points  perpendiculars  or  ordinates  are  erected,  and  upon  them  are  marked 


TOPOGRAPHICAL   DRAWING.  365 

the  heights  of  the  ground  above  the  base,  and  the  marks  are  joined  by 
straight  lines.  To  express  rock  in  a  cut,  it  is  generally  represented  by  di- 
agonal lines;  rivers  are  represented  in  section  by  cross  lines  or 'colored  in 
blue ;  the  depth  of  the  sounding  in  a  mud  bottom  by  masses  of  dots. 

Since  it  would  be  in  general  impossible  to  express  the  variations  of  the 
surface  of  the  ground  in  the  same  scale  as  that  adopted  for  the  plan,  it  is 
usual  therefore  to  make  the  vertical  scale  larger  than  that  of  the  horizontal 
lines  one,  in  proportion  of  10  or  20  to  1.  Thus,  if  the  horizontal  scale  of 
the  plan  be  400  feet  to  the  inch,  the  vertical  scale  would  be  40  or  20  feet 
to  the  inch. 

For  the  purpose  of  facilitating  the  plotting  of  profiles,  profile  paper  is 
prepared,  on  which  are  printed  horizontal  and  vertical  lines ;  the  horizon- 
tal lines  being  ruled  at  a  distance  of  gV  of  an  inch  from  each  other,  every 
fifth  line  being  coarser,  and  every  twenty-fifth  still  heavier  than  the  others. 
Each  of  the  spaces  is  usually  considered  one  foot.  The  vertical  lines  are 
one  quarter  of  an  inch  distant  from  each  other,  every  tenth  line  being 
made  more  prominent  than  the  others ;  these  spaces  in  general  represent 
a  distance  of  100  feet,  the  usual  distance  between  stations  on  a  railroad. 
Much  time  is  saved  by  the  use  of  this  paper,  both  in  plotting,  and  in  read- 
ing the  measurements  after  they  are  plotted. 

In  the  plotting  of  sections  afcross  the  line,  which  are  extended  but  little 
beyond  the  line  of  the  cut  or  embankment,  equal  vertical  and  horizon- 
tal scales  are  adopted ;  these  plots  are  mostly  to  determine  the  position  of 
the  slope,  or  to  assist  in  calculating  the  excavation.  To  facilitate  these, 
cross  section  paper  is  prepared,  ruled  with  vertical  and  horizontal  lines, 
forming  squares  of  TV  of  an  inch  each.  Every  fifth  line  in  each  direction 
is  made  prominent.  "When  cross  sections  are  extended  to  show  the  grade  of 
cross  road,  or  changes  of  level  at  considerable  distance  from  the  line  of 
rail,  the  same  scales  vertical  and  horizontal  are  adopted  as  in  the  longitu- 
dinal section  or  profile. 

It  will  be  observed  in  fig.  34,  that  the  upper  or  heavy  line  represents 
the  line  of  the  rail,  the  grades  being  written  above ;  this  is  the  more  usual 
way,  but  sometimes,  as  in  fig.  36,  the  profile  and  plan  are  combined ;  that 


is  the  heights  and  depths  above  and  below  the  grade  line  of  the  road  are 


366  TOPOGRAPHICAL   DRAWING. 

transferred  to  the  plan,  and  referred  to  the  line  in  plan,  which  becomes 
thus  a  representation  both  in  plan  and  elevation. 

Cross  sections,  for  grades  of  cross  roads,  etc.,  are  usually  plotted  be- 
neath or  above  the  profile ;  they  may,  if  necessary,  be  plotted  across  the 
line  when  plan  and  profile  are  combined. 

Besides  the  complete  plans  as  above,  giving  the  details  of  the  location, 
land  plans,  so  called,  are  required,  showing  the  position  and  direction  of 
all  lines,  of  fences  and  boundaries  of  estates,  with  but  very  few  of  the 
topographical  features.  The  centre  line  of  road  is  represented  in  bold 
line,  and  at  each  side,  often  in  red,  are  represented  the  boundaries  re- 
quired for  the  purposes  of  way.  In  general,  a  width  of  five  rods  is  the 
amount  of  land  set  off,  lines  parallel  to  the  central  line  being  at  a  distance 
of  two  and  one  half  rods  on  each  side ;  but  when,  owing  to  the  depth  of  the 
cut  or  embankment,  the  slopes  run  out  beyond  this  limit,  the  extent  is  de- 
termined by  plotting  a  cross  section  and  transferring  the  distances  thus 
found  to  the  plan,  and  enclosing  all  such  points  somewhat  within  the 
limits  as  set  off  for  railway  purposes.  These  plans  are  generally  filed  in 
the  register's  office  for  the  county  through  which  the  line  passes. 

For  Railway  plans  prepared  for  the  English  Parliament  certain  regulations  are  defined. 

The  plan  must  be  upon  a  scale  of  at  least  four  inches  to  a  mile,  and  must  describe  the 
line  or  situation  of  the  whole  of  the  proposed  work,  and  the  lands  through  which  the 
same  will  be  made,  and  also  any  communication  to  be  made  with  the  proposed  work.  If 
the  plan  is  on  a  scale  less  than  }  of  an  inch  to  every  100  feet,  there  must  be  an  additional 
plan  upon  that  scale  (1 :  4,800)  of  any  building,  yard,  and  of  any  ground  cultivated  as  a 
garden. 

The  plan  to  exhibit  thereon  the  distances  in  miles  and  furlongs,  from  one  of  the  ter- 
mini, with  memoranda  of  the  radii  of  all  curves  less  than  one  mile  in  length,  noted  on  the 
plan  in  chains  where  the  curve  occurs.  When  a  tunnel  is  intern1  :-u  to  be  constructed,  it 
must  be  marked  in  by  a  dotted  line  on  the  plan.  • 

Each  distinct  property,  divided  by  any  visible  boundary  from  another  property,  should 
have  a  separate  number ;  with  this  exception,  that  any  collection  of  buildings  and  grounds 
within  the  curtilage  of  R  building,  belonging  to  one  person  and  in  one  occupation,  may  be 
tlescril  C'l  under  one  m; ,  V r ;  thus,  farm  house,  yn  '  &r .  TVhcn  it  is  necessary  to  inter- 
pose a  ruirnber,  a  duplic-^  number  should  be  addei  thi1  ••  f><>.  ?>a.  'he  r.umbei.-!  ig  should 
recommence  in  every  parish. 

All  lands  included  within  the  limit  of  deviation,  shown  by  lines  drawn  on  plan,  and 
all  lands  which  those  lines  touch,  must  be  numbered  and  described.  Public  roads,  and 
private  roads  if  fenced  out,  should  have  a  separate  number.  Navigable  and  mill  streams 
.  aratel; 


It  is  sometimes  usual,  at  the  c^mple+ion  of  a  railway,  to  make  plans  of 
the  works  as  finished ;  and,  if  a  profile  of  the  line,  to  represent  the  differ- 


TOPOGRAPHICAL   DRAWING. 


367 


ent  strata  or  rocks  in  the  cuts,  with  their  dip  or  inclination.  This  is  more 
properly  a  geological  profile ;  the  different  rocks  are  usually  distinguished 
by  different  colors  and  explained  by  marginal  notes  and  squares,  some- 
times by  marks,  dots,  and  cross  hatchings,  as  fig.  37,  often  by  color  in  ad- 


dition. Figs  I.,  II.,  III.,  IY.,  represent  the  primary,  secondary,  tertiary, 
and  recent  plutonic  rocks.  Fig.  1  represents  the  primary  fossiliferous 
strata ;  figs.  2,  3,  4,  the  secondary,  tertiary,  and  recent  strata.  In  the  plots 
of  geological  sections,  it  is  especially  requisite  that  the  different  strata 
should  be  accurately  represented. 

In  plotting  hydrometrical  or  marine  surveys,  the  depths  of  soundings 
are  not  expressed  by  sections,  but  by  figures  written  on  the  plan,  express- 
ing the  sounding  or  depth  below  a  datum  line,  generally  that  of  high 
water.  The  low  water  line  is  generally  represented  by  a  single  continued 
line.  The  soundings  are  generally  expressed  in  fathoms,  sometimes  in  feet. 

It  is  usual,  in  plotting  from  a  field  book,  to  make  first  but  a  rough 
draft,  and  then  naak^  a  fin'  hed  copy  on  another  sheet.  In  tL.  first,  many 
lines  of  construction,  balances  of  survey,  and  trial  lines  are  drawn  which 
are  unnecessary  in  the  copy ;  outlines  of  natural  features  are  sketched 
roughly,  but  the  plotting  of  surveys,  and  such  lines  and  points  as  are  to 
be  preserved  in  the  copy,  must  be  ])]<•  I  '  1i  accuracy. 

Tho  "i  .m-  v.-ay  of  transferri . .  topy,  is  by  snperpo- 

fc'aon  o.  the  plan  above  the  sheet  intended  for  the  copy,  and  pricking 
through  every  intersection  of  lines  on  the  plan  and  all  such  points  as  may 
be  necessary  to  preserve.  The  clean  paper  should  be  laid  and  fastened 
smoothly  on  the  drawing  board,  the  rough  draft  should  be  laid  on  smoothly 
and  retained  in  it»  position  by  weig  !'s.  The  needle  must 

be  held  perpenulcular  to  the  surface  of  the  plan  and  pressed  through  both 
sheets ;  begin  at  one  side  and  \\  rk  with  ~ys*em,  so  as  not  to  prick  through 
each  point  but  once,  nor  omit  any ;  make  the  important  points  a  trifle  the 


368  TOPOGRAPHICAL  DE AWING. 

largest.  For  the  irregular  curves,  as  of  rivers,  make  frequent  points,  but 
very  small  ones.  On  removing  the  plan  select  the  important  points,  those 
denning  leading  lines ;  draw  in  these,  and  the  other  points  will  be  easily 
recognized  from  their  relative  position  to  these  lines.  "When  any  point 
has  not  been  pricked  through,  its  place  may  be  determined  by  taking  any 
two  established  points  adjacent  to  the  cjne  required,  and  with  radii  equal  to 
their  distance,  on  the  plan,  from  the  point  required,  describing  arcs,  on 
the  copy,  on  the  same  side  of  the  two  points,  their  intersection,  will  be  the 
point  desired.  In  this  way,  as  in  a  trigonometrical  survey,  having  estab- 
lished the  two  extremes  of  a  base,  a  whole  plan  may  be  copied.  For  this 
purpose  the  triangular  -compasses  (p.  27)  will  be  found  very  convenient. 
In  extensive  drawings  it  is  very  common  to  prick  off  but  a  few  of  the 
salient  points,  and  fill  in  by  intersections,  as  above,  or  by  copying  detached 
portions  on  tracing  paper,  and  transferring  them  to  the  copy ;  the  position 
of  each  sketch  being  determined  by  the  points  pricked  off,  the  transfer  is 
made  by  pricking  through  as  above,  or  by  transfer  paper  placed  between 
the  tracing  and  the  copy. 

Tracing  paper  is  a  thin,  transparent  paper,  prepared  expressly  for  the 
purpose  of  making  copies  of  drawings.  Placed  above  the  drawing,  every 
line  shows  through,  and  is  traced  directly  with  the  pen,  in  India  ink. 
These  tracings  are  used  mostly  to  preserve  duplicates  of  finished  drawings. 
As  tracing  paper  is  of  too  slight  a  texture  to  bear  much  handling,  cotton 
cloth  is  prepared  and  sold  under  the  name  of  "  vellum  tracing  paper." 
When  it  is  necessary  to  use  tracing  paper  drawings  to  work  by,  it  is  usual 
to  attach  them  to  sheets  of  white  paper,  which  serves  both  to  bring  out 
the  lines  and  to  strengthen  the  paper. 

Duplicates  of  drawings  are  now  very  neatly  executed,  and  of  course 
accurately  copied,  by  the  Photographic  process,  but  it  is  more  applied  to 
mechanical  and  architectural  drawings  than  topographical. 

An  accurate  and  rapid  way  of  tracing,  on  drawing  paper,  plans  of  small 
extent,  is  by  means  of  an  instrument  called  a  copying  glass.  It  consists 
of  a  large  piece  of  plate  glass  set  in  a  frame  of  wood,  which  can  be  in- 
clined at  any  angle  in  the  same  manner  as  a  reading  or  music  desk.  On 
this  glass  is  first  laid  the  original  plan,  and  above,  the  fair  sheet,  and  the 
frame  being  raised  to  a  suitable  angle,  a  strong  light  is  thrown  by  reflect- 
ors or  otherwise  on  the  under  side  of  the  glass,  whereby  every  line  in  the 
original  plan  is  seen  distinctly  through  the  fair  sheet,  and  the  copy  is  made 
at  once,  in  ink,  as  on  tracing  paper,  and  finished  while  being  traced.  This 
same  process,  on  a  small  scale,  is  adopted  by  putting  the  plans  upon  a 
pane  of  glass  in  a  window. 


TOPOGRAPHICAL  DRAWING.  369 

Plans  of  great  extent  cannot  be  conveniently  copied  by  means  of  the 
copying  glass.  Moreover,  being  often  mounted  on  cloth,  which  renders 
them  opaque,  they  do  not  admit  of  being  traced  in  this  way.  In  such 
cases  the  copy  may  be  made  by  means  of  transfer  paper.  The  plan  is  first 
traced  in  ink  on  tracing  paper  or  cloth,  black  leaded  or  transfer  paper  is 
then  placed  on  the  fair  sheet,  and  the  tracing  paper  copy  is  placed  above. 
All  is  steadied  by  numerous  weights  along  the  edges,  or  by  drawing  pins 
fixed  into  the  drawing  board.  A  fine  and  smooth  point  is  then  passed 
over  each  boundary  or  mark  on  the  tracing  with  a  pressure  of  the  hand 
sufficient  to  cause  a  clear,  pencilled  mark  to  be  left  on  the  fair  sheet  by 
the  black  leaded  or  transfer  paper.  The  whole  outline  is  thus  obtained, 
and  afterwards  drawn  in  ink.  The  copyist  should  be  careful  in  his  manipu- 
lations, so  as  not  to  transfer  any  other  lines  than  those  required,  nor  leave 
smutches  on  the  fair  sheet. 

Plans  may  be  copied  on  a  reduced  or  enlarged  scale  by  means  6"f  the 
pentagraph,  but  the  more  usual  way  is  by  means  of  squares.  Draw  on 
the  plan  to  be  reduced,  a  series  of  parallel  and  equi-distant  lines,  with 
others  perpendicular  to  them  at  similar  intervals,  thus  covering  the  whole 
surface  with  equal  squares.  On  the  clean  sheet  draw  a  similar  set  of 
squares,  but  with  their  sides  to  the  desired  reduced  scale ;  one-half,  one- 
third,  &c.,  as  the  case  may  be.  Then  copy  into  each  small  square  all  the 
points  and  lines  in  the  corresponding  square  on  the  plan,  in  their  true  po- 
sition relative  to  the  sides  and  corners  of  the  square,  reducing  each 
distance,  by  proportional  dividers,  or  by  eye  as  may  be  necessary,  in  the 
given  ratio.  In  reducing  by  the  camera  lucida,  squares  on  the  plan  are 
brought  apparently  to  coincide  with  squares  in  the  copy,  and  the  details 
as  seen  through  the  prism  of  the  instrument,  are  then  filled  in  with  the 
pencil.  This  instrument  is  used  in  the  U.  S.  Coast  Survey  office,  but  it 
cannot  reduce  smaller  than  one-fourth  without  losing  distinctness,  and  it  is 
very  trying  to  the  eyes. 

Finishing  the  Plan  or  Map.— In  general,  in  topographical  as  in  ar- 
chitectural and  mechanical  drawings,  the  light  is  supposed  to  fall  upon  the 
surface  in  a  diagonal  direction  from  the  upper,  left-hand  corner.  This  rule 
is  not  uniform :  by  some  draughtsmen  the  light  is  introduced  at  the  lower 
left,  and  hills  are  mostly  represented  under  a  vertical  light,  although  the 
oblique  adds  more  to  the  picturesque  effect.  The  plan  is  usually  so  drawn 
that  the  top  may  represent  the  north,  and  the  upper  left-hand  corner  is 
then  the  north-west. 

In  inking  in,  commence  first  with  the  light  lines,  since  a  mistake  in 
these  lines  may  be  covered  by  the  shade  lines.  Describe  all  curves  which 
24 


3<0  TOPOGRAPHICAL   DRAWING. 

are  to  be  drawn  with  compasses  or  sweeps  before  the  straight  lines,  for  it 
is  easier  to  join  neatly  a  straight  line  to  a  curve  than  the  opposite.  Ink 
in  with  system,  commencing  say  at  the  top ;  ink  in  all  light  lines  running 
easterly  and  westerly,  then  all  light  lines  running  northerly  and  southerly, 
then  commence  in  the  same  way  and  draw  in  the  shade  lines.  It  will  of 
course  be  understood  that  elevated  objects  have  their  southern  and  eastern 
outline  shaded,  whilst  depressions  have  the  northern  and  western ;  thus 
in  conventional  signs  roads  are  shaded  the  opposite  to  canals.  Having 
inked  in  all  lines  that  are  drawn  with  a  ruler  or  described  with  compasses, 
commence  again  at  one  corner  to  fill  in  the  detail,  keeping  all  the  rest  of 
the  plan  except  what  you  are  actually  at  work  upon  covered  with  paper, 
to  protect  it  from  being  soiled.  The  curved  lines  of  brooks,  fences,  &c., 
are  sometimes  drawn  with  a  drawing  pen,  sometimes  with  a  steel  pen  or 
goose  quill.  The  latter  are  generally  used  in  drawing  the  vertical  lines 
of  hills. 

Boundary  lines  of  private  properties,  of  townships,  of  counties,  of 
states,  <fcc*.,  are  indicated  by  various  combinations  of  short  lines,  dots  and 
crosses,  thus : 


All  plans  should  have  meridian  lines  drawn  on  them,  also  scales.  Plate 
XCI.  shows  some  designs  for  meridians.  In  these  diagrams  it  will  be 
observed  that  both  true  and  magnetic  meridians  are  drawn ;  this  is  desira- 
ble ,when  the  variation  is  known,  but  in  many  surveys  merely  the  magnet- 
ic meridian  is  taken ;  in  these  cases  this  line  is  simply  represented  with 
half  of  the  barb  of  the  arrow  at  the  north  point,  and  on  the  opposite  side 
of  the  line  from  the  tme  rneridan.  Scales  are  drawn  or  represented  in  va- 
rious forms,  as  may  be  seen  in  the  following  plates,  or  the  proportion  of 
the  plan  to  the  ground  is  expressed  decimally,  as  the  number  of  feet, 
chains,  etc.,  to  the  inch. 

Lettering. — The  style  in  which  this  is  done  very  much  affects  the  gen- 
eral appearance  of  the  plan.  Great  care  must  be  taken  in  the  selection 
and  character  of  the  type,  and  in  the  execution.  The  usual  letters  are  the 

R  o  M  A  x . 

ABCDEFGHIJKLMNO 
PQRSTUVWXYZA 


TOPOGRAPHICAL   DRAWING.  371 


SMALL     ROMAN. 


abcdefghijklmnopqrstuv 

wxyz,;:. 

1234567890 


ITALIC. 


ABCDEFGHIJKL 
MNQPQRSTUYW 


abcdefghijklmnopqrst 
uvwxyz;: 


GOTHIC,     OR     EGYPTIAN. 


ABCDEFCHIJKLMN 
OPQRSTUVWX 

¥  £•£&      •  • 


372  TOPOGRAPHICAL  DRAWING. 


ABCDEGHJKLMORSTUWY 

abcdefghijklmnopqrstuvwxy 


a  fe  ©  €  @  t  g  1  1  m  ©  p  E>  s  tm  w 

ABGDE&HJKLMORSTTJWXY 

ABCDEFGHIJKLiNOPQRSTDfim 

ABCDEGHJKMORSTUWXY 

abcdefghijklmnopqrstuvwxyz 


TOPOGRAPHICAL   DRAWING. 
OLD     ENGLISH. 


373 


OLD     ENGLISH     SCRIBE     BLACK. 


GERMAN     TEXT. 


374r  TOPOGRAPHICAL   DRAWING. 


ENGLISH    OUTLINE.    , 


9  c  D  a 

PI.  XCII.  represents  a  mechanical  method  of  constructing  letters  and 
figures.  This  plate  should  be  copied  by  the  draftsman,  and  on  a  much 
larger  scale,  by  drawing  first  the  system  of  squares  or  parallelograms,  and 
then  sketching  in  the  letters ;  in  this  way  well  formed  and  proportioned 
letters  can  always  be  made,  and  from  a  selection  of  alphabets  the  lettering 
may  be  selected  and  transferred  to  the  plan. 

PL  XCIII.  are  examples  of  titles,  intended  merely  as  an  illustration  of 
the  form  of  letters  and  their  arrangement,  the  scale  being  much  smaller 
than  that  used  on  plans,  except  such  as  are  drawn  to  a  very  small  scale. 
It  will  be  observed  that  the  more  important  words  are  made  in  promi- 
nent type.  The  lower  part  of  the  title  should  always  contain,  in  small 
character,  the  name  of  the  party  making  the  survey,  and  also  the  name 
of  the  draftsman,  with  date  of  the  execution  of  the  plan :  if  the  survey 
was  made  some  time  previous,  the  date  of  the  survey  should  be  given. 
If  the  plan  is  compiled  from  several  surveys,  the  authorities  should,  if 
possible,  be  given.  The  lettering  of  the  title  should  be  in  lines  parallel  to 
the  bottom  of  the  plan,  and,  in  general,  the  great  mass  of  lettering  in  the 
body  of  the  plan  is  formed  in  similar  lines ;  but  curved  lines  are  often  not 
only  essential,  but  they  materially  contribute  to  the  beauty  of  the  plan. 
Thus  on  crooked  boundaries  on  outlines  of  maps,  the  lettering  should  fol- 
low the  general  curve  of  the  boundary;  also  on  crooked  rivers,  lakes, 
seas,  &c. ;  on  irregular  or  straggling  pieces  of  land,  in  order  to  show  the 
extent,  connection,  or  proprietorship  thereof,  the  lettering  should  follow 
the  central  line  of  such  a  tract ;  and  if  pieces  of  land  be  very  oblong  in 
form  but  regular  in  outline,  the  lettering  will  be  central  in  the  direction  of 
the  longest  side.  The  lettering  of  roads,  streets,  &c.,  is  always  in  the  di- 
rection of  the  line  of  road.  Curved  lines  of  lettering  are  often  introduced 
into  extended  titles  to  take  off  the  monotonous  appearance  presented  by  a 
great  number  of  straight  lines  of  writing. 

The  direction  of  all  lettering  should  be  so  as  to  be  read  from  left  to 
right.  If  shades  or  shadows  are  introduced  to  give  relief  or  break  up  the 
monotony,  they  should  be  uniform  with  the  rest  of  the  plan. 

On  the  Spacing  of  Letters. — It  will  be  observed  that  letters  vary  very 
considerably  in  their  width,  the  I  being  the  narrowest,  and  the  TFthe 


TOPOGRAPHICAL   DRAWING. 


375 


widest :  if  therefore  the  letters  composing  a  word  be  spaced  off  at  equal 
distances  from  centre  to  centre,  the  interval  or  space  between  the  letters 
will  be  more  in  some  cases  than  in  others.  Tims,  in  the  word 

R    A    I    L  W  A  Y 

To  avoid  this,  write  in  first  one  letter,  and  then  space  off  a  proper  interval, 
and  then  write  in  the  next  letter,  and  then  space  off  the  interval  as  before, 
and  so  on,  thus, 

RAILWAY 

When,  as  frequently  happens,  the  words  are  very  much  extended,  in  order 
to  embrace  and  explain  a  large  extent  of  surface  or  boundary,  and  the 
space  occupied  by  the  letter  is  small  in  comparison  with  the  interval,  the 
disparity  of  intervals  will  not  be  noticed,  and  the  letters  may  be  then  laid 
off  at  equal  spaces  from  centre  to  centre,  thus : 


A 


B 


W 


When  the  lines  of  lettering  are  curved,  the  same  rules  for  spacing  are  to 
be  observed  as  above.  If  the  letters  are  upright,  as  Eoman  or  Gothic,  the 
sides  of  each  letter  are  to  be  parallel  to  the  radius  drawn  to  the  centre  of 
the  letter,  and  the  bottom  and  top  lines  at  right  angles  to  it.  If  the  let- 
ters be  inclined  as  Italic  letters,  then  the  side  lines  of  the  letters  must  be 
inclined  to  the  central  radial  line,  as  on  a  horizontal  line  they  are  inclined 
to  the  perpendicular. 


In  laying  off  letters  by  equal  intervals,  it  is  usual  to  count  the  number 
of  letters  in  the  word,  and  fix  the  position  on  the  plan  of  the  central  one, 
and  then  space  off  on  each  side :  this  is  particularly  important  in  titles, 
when  it  is  necessary  that  many  lines  should  have  their  extremities  at  uni- 


376  TOPOGRAPHICAL   DRAWING. 

form  distances  from  the  centre  line.  In  laying  off  the  title,  we  determine 
what  is  necessary  to  be  included  in  the  title,  the  space  it  must  occupy,  the 
number  of  lines  necessary,  and  the  style  and  arrangement  of  characters  to 
be  used.  Thus,  if  the  title  were,  plan  of  a  proposed  terminus  of  the  Har- 
lem Railroad  at  New  York,  1857,  knowing  the  space  to  be  occupied,  we 
can  write  the  title  thus : — 


of 


We  now  draw  parallel  lines  at  intervals  suited  to  the  character  of  the  type 
we  intend  to  employ  for  the  different  words.  Harlem  Railroad  is  the  line 
to  be  made  most  prominent  ;  this,  calling  the  interval  between  the  words 
one  letter,  includes  15  letters  ;  or,  if  we  consider  /,  with  its  proper  interval, 
but  half  a  letter,  (which  will  be  found  a  very  good  rule  in  spacing,)  141  ; 
hence  the  centre  of  the  line  will  be  7^  letters  from  the  beginning,  or  \  of 
the  space  occupied  by  the  letter  H  and  its  interval.  Draw  a  perpendicu- 
lar line  at  the  centre,  and  write  in  R  in  such  a  character  as  may  suit  the 
position  to  be  filled,  and  lay  off  by  letters  and  spaces  the  other  letters. 
The  line  Harlem  Railroad  is  intended  to  occupy  the  whole  length  of 
space  ;  that  is,  it  must  be  the  longest  line  in  the  title,  and  the  lines  above 
and  below  must  gradually  diminish,  forming  a  sort  of  double  pyramid. 
Proposed  Terminus  includes  16i  letters,  the  I  and  interval  between  the 
words  being  rated  as  above,  we  find  the  centre  to  be  nearly  midway  be- 
ween  the  words.  These  words  including  more  letters,  and  being  confined 
within  less  space,  must  be  in  smaller  character  than  the  preceding  ;  and  as 
a  further  distinction,  a  different  style  should  be  adopted.  Having  deter- 
mined this,  we  proceed  to  write  in  the  letters  as  before,  and  in  the  same 
way  with  the  other  lines,  the  prepositions  as  unimportant  are  always  writ- 
ten in  small  type. 


TOPOGRAPHICAL   DRAWING.  377 


jof  the. 


HARLEM  RAILROAD 


.at. 


NEW    YORK 

1857_ 

I 

In  general  it  is  better  that  letters  should  be  first  written  on  a  piece  of 
paper,  distinct  from  the  plan,  as  repeated  trials  may  be  necessary  before  one 
is  arranged  to  suit  the  draughtsman.  Having  formed  a  model  title,  it  may  be 
copied  in  the  plan  by  measures  or  by  tracing  and  transfer  paper.  There 
are  some  words,  such  as  Plan,  Map,  Section,  Scale,  Elevation,  &c.,  which,  as 
they  are  of  constant  occurrence,  may  be  cut  in  stencil ;  sometimes  whole 
alphabets  are  thus  cut  and  words  compounded.  It  will  be  found  very  con- 
venient for  a  draughtsman  if  he  makes  tracing  or  copies  of  such  titles  as 
he  meets  with,  and  preserves  them  as  models  ;  for  there  is  no  manipulation 
on  a  plan  that  contributes  more  to  the  effect  than  good  lettering  and  arrange- 
ment of  titles,  and  considerable  practice  should  be  expended  in  acquiring 
a  facility  in  lettering,  and  for  the  first  start,  perhaps  nothing  will  be  found 
more  valuable  than  tracing  good  examples. 

We  have  treated  of  mechanical  methods  by  which  most  persons  can 
learn  to  form  letters  and  words ;  but  it  must  be  borne  in  mind  that  the 
distances  between  letters  on  the  plan  are  only  intended  to  suit  the  eye ; 
if  therefore  a  person  accustom  himself  to  spacing,  so  that  his  eye  is  cor- 
rect, there  will  be  no  necessity  of  laying  off  by  dividers;  in  this  mode, 
such  letters  as  A  and  V,  L  and  T  are  brought  nearer  each  other  than  the 
regular  interval.  In  general  it  may  be  observed  in  reference  to  to  the 
lettering  of  Topographical  Drawings,  stiff  letters  like  those  of  stencil 
should  not  be  introduced,  but  there  should  be  such  variety,  incident  on 
construction  by  the  pen,  as  may  be  consonant  with  the  rest  of  the  drawing. 


378  TOPOGRAPHICAL   DRAWING. 


TINTED     TOPOGRAPHICAL     DRAWING. 

We  have  hitherto  treated  of  the  representation  of  the  features  of  the 
country  by  the  pen  only,  but  it  may  be  done  full  as  effectively  and  much 
more  expeditiously  by  means  of  the  brush  and  water  colors,  either  by 
India  ink  alone,  or  by  various  tints,  or  by  the  union  of  both. 

The  most  important  colors  for  conventional  tints  are,  (besides  India 
ink),  Indigo  (blue),  Carmine  (or  crimson  lake),  and  Gamboge  (yellow), 
used  separately  or  compounded.  Besides  these,  Burnt  Sienna,  Yellow 
Ochre,  and  Vermilion  are  sometimes  used,  although  the  three  first  are 
susceptible  of  the  best  combinations,  and  the  others  are  generally  used 
alone. 

The  following  conventional  colors  are  used  by  the  French  Military  En- 
gineers in  their  colored  topography.  "Woods,  yellow  •  using  gamboge  and 
a  very  little  indigo.  Grass  land,  green ;  made  of  gamboge  and  indigo. 
Cultivated  land,  brown y  lake,  gamboge,  and  a  little  India  ink;  "Burnt 
Sienna  "  will  answer.  Adjoining  fields  should  be  slightly  varied  in  tint. 
Sometimes  furrows  are  indicated  by  strips  of  various  colors.  Gardens  are 
represented  by  small  rectangular  patches  of  brighter  green  and  Thrown. 
Uncultivated  land,  marbled  green  and  light  'brown.  Brush,  brambles,  &c., 
marbled  green  and  yellow.  Heath,  furze,  &c.,  marbled  green  and  pink. 
Vineyards,  purple;  lake  and  indigo.  Sands,  a  light  "brawn ;  gamboge 
and  lake  ;  "  Yellow  Ochre  "  will  do.  Lakes  and  rivers,  light  Hue,  with  a 
darker  tint  on  their  upper  and  left  hand  sides.  Seas,  dark  blue,  with  a  lit- 
tle yellow  added.  Marshes,  the  Hue  of  water,  with  spots  of  grass  green, 
the  touches  all  lying  horizontally.  Roads,  brown  •  between  the  tints  for 
sand  and  cultivated  ground,  with  more  India  ink.  Hills,  greenish  brown ; 
gamboge,  indigo,  lake  and  India  ink.  Woods  may  be  finished  up  by 
drawing  the  trees  and  coloring  them  green,  with  touches  of  gamboge  to- 
wards the  light,  (the  upper  and  left  hand  side,)  and  of  indigo  on  the  op- 
posite side. 

In  addition  to  the  conventional  colo*rs,  a  sort  of  imitation  of  the  con- 
ventional signs  already  explained  are  introduced  in  color  with  the  brush, 
and  shadows  are  almost  invariably  introduced.  The  light  is  supposed  to 
come  from  the  upper  left  hand  corner,  and  to  fall  nearly  vertical,  but  suf- 
ficiently oblique  to  allow  of  a  decided  light  and  shade  to  the  slopes  of 
hills,  trees,  &c.  The  shadow  of  any  object  will  therefore  surround  its 


TOPOGRAPHICAL     DRAWING.  379 

lower  right  hand  outlines.  After  the  shadow  has  been  painted,  the  out- 
line of  the  object  is  strengthened  by  a  heavy  black  line  on  the  side  oppo- 
site the  light.  The  flat  tints  are  first  laid  on  as  above,  and  then  the  con- 
ventional signs  are  drawn  in  with  a  pencil  and  colored  in  with  appropriate 
and  more  intense  tints ;  the  shadows  are  generally  represented  in  India  ink. 

Hills  are  shaded,  not  as  they  would  appear  in  nature,  but  on  the  con- 
ventional system  of  making  the  slopes  darker  in  proportion  to  their  steep- 
ness :  the  summit  of  the  highest  ranges  being  left  white.  This  arrange- 
ment, though  obviously  incorrect  in  theory,  has  the  advantage  of  being 
generally  understood  by  those  not  accustomed  to  plan  drawing,  and  is 
also  easy  of  execution.  Wash  the  surface  first  with  the  proper  flat 
tint,  trace  in  with  a  pencil,  outlines ;  then  lay  on  in  India  ink  tints  pro- 
portioned in  intensity  to  the  height  of  the  hills  and  steepness  of  the 
slopes.  To  soften  the  tints  two  brushes  are  used,  one  as  a  color  brush,  the 
other  as  a  water  brush  :  the  tints  are  laid  on  with  the  first,  and  softened 
by  passing  the  water  brush  rapidly  along  the  edges.  The  water  brush 
must  not  have  too  much  water,  as  it  would  in  that  case,  lighten  the  tint 
to  a  greater  extent  than  is  intended,  and  leave  a  ragged  harsh  edge.  Tints 
may  be  applied  in  very  light  shades,  one  tint  over  another,  with  the 
boundary  of  the  upper  tint  not  reaching  the  extreme  limit  of  the  tint 
below  it.  When  depth  of  shade  is  required,  it  is  best  produced  by  appli- 
cation of  several  light  tints  in  succession :  no  tint  is  to  be  laid  over  the 
other  until  the  first  is  dry,  and  a  little  indigo  mixed  with  the  India  ink  im- 
proves its  color  and  adds  to  the  richness  of  effect. 

When  woods  have  to  be  represented,  the  shading  used  for  the  trees 
instead  of  interfering  with  the  shadows  due  to  the  slopes,  may  be  made  to 
harmonize  with  them,  and  contribute  to  the  general  effect  by  presenting 
greater  or  less  depth,  according  to  the  position  of  the  woods  on  the  sides, 
or  summits  of  the  hills. 

An  expeditious  and  effective  way  of  representing  hills  with  brush,  a 
species  of  imitation  of  hills  drawn  with  a  pen  on  the  vertical  system,  is 
effected  by  pressing  out  flat  the  brush  to  a  sort  of  comb-like  edge ;  draw- 
ing this  over  a  nearly  dry  surface  of  India  ink,  and  then  brushing  lightly 
or  more  heavily  between  the  contours,  according  to  the  steepness  of  the 
slope,  each  of  the  comb-like  teeth  making  its  mark. 

Kivers  and  masses  of  water  may  be  shaded  in  with  a  color  and  water 
brush  as  above,  or  by  superposition  of  light  tints,  a  shadow  may  be 
thrown  from  the  bank  towards  the  light,  and  the  outline  of  this  bank 
strengthened  with  a  heavy  black  line.  The  tints  are  to  be  in  indigo,  the 
shadows  in  India  ink. 


380 


TOPOGRAPHICAL   DEAWING. 


Topographical  drawings  may  be  made  in  water  color  with  but  one  tint,  as 
India  ink,  or  ink  mixed  with  a  little  sepia.  The  conventional  signs  are  in 
imitation  of  pen  drawings,  the  hills  in  softened  tint,  or  drawn  with  the 
comb-edged  brush,  and  the  rivers  shaded  with  superposed  tints. 

Most  artistic  and  effective  drawings  are  made  of  hills  as  they  would 
appear  in  nature,  under  an  oblique  light :  the  sides  of  the  hills  next  the 
light  receiving  it  more  or  less  brilliantly,  according  as  they  are  inclined 
more  or  less  at  right  angles  with  its  rays,  and  the  shades  on  the  sides  re- 
moved from  the  light  increasing  in  intensity  as  the  slopes  increase  in  steep- 
ness. This  style  may  be  rendered  most  expressive  by  a  skilful  draughts- 
man, especially  when  the  character  and  strike  of  the  hills  are  favorable  to 
the  direction  of  the  light,  but  with  this  style  of  representation  the  hills  are 
generally  made  to  partake  more  or  less  of  the  same  character,  appearing 
almost  uniformly  steepest  on  the  sides  removed  from  the  light.  It  partakes 
therefore  more  of  the  artistic  character,  more  difficult  to  execute,  and 
conveying  information  in  a  more  vague  manner  than  by  the  common  to- 
pographical conventionalities. 


TOPOGEAPHICAL   DRAWING.  381 

In  preparing  the  paper  for  a  tinted  drawing  it  must  be  damp-stretched 
upon  the  drawing  board  in  such  a  manner  that  the  moisture  of  the  color 
will  not  cause  undulations  or  blisters  on  the  surface :  this  process  is  pre- 
viously, described  at  page  37.  Having  prepared  a  sheet  of  paper  accord- 
ing to  the  directions  there  given,  first  draw  in  the  lines  in  pencil,  and  af- 
terwards repeat  them  with  a  very  light  ink  line :  a  soft  sponge  well  satu- 
rated should  then  be  passed  quickly  over  the  surface  of  the  drawing,  in 
order  to  remove  any  portions  of  the  ink  which  would  be  liable  to  mix  with 
the  tint  and  mar  its  uniformity.  When  the  paper  is  dry  proceed  to  lay  on 
the  conventional  tints. 

Great  care  is  necessary  in  preparing  and  combining  the  diiferent  colors, 
and  attention  to  certain  mechanical  conditions  and  rules  must  be  observed 
in  order  to  insure  neatness  and  despatch  in  execution.  The  cakes  of  color 
are  quite  brittle,  and  it  is  well  to  moisten  the  end  and  allow  it  to  soften 
slightly  before  using,  then  rub  upon  a  perfectly  clean  palette,  with  a  few 
drops  of  pure  water,  a  sufficient  quantity  of  color  to  tinge  to  the  proper 
intensity  as  much  water  as  will  be  required  for  the  whole  drawing,  This 
should  be  thoroughly  mixed  with  the  brush,  and  as  often  as  the  brush  is 
filled,  to  insure  uniformity  in  the  tint. 

Previous  to  applying  the  tint,  it  is  well  to  moisten  the  surface  to  be  col- 
ored with  clean  water,  which  will  prevent  the  tint  from  drying  too  rapidly 
at  the  edges.  In  tinting  never  allow  the  edge  to  dry  until  the  whole  sur- 
face is  covered:  leave  a  little  superfluous  color  along  the  edge  whilst 
filling  the  brush.  Great  caution  is  necessary  in  approaching  the  outlines 
of  the  drawing,  and  the  point  of  the  brush  should  be  used  so  as  not  to 
overrun  the  lines. 

In  applying  a  flat  tint  to  large  surfaces,  let  the  drawing  board  be  in- 
clined upwards  at  an  angle  of  5  or  6  degrees,  so  as  to  allow  the  color  to 
flow  downwards  over  the  surface.  With  a  moderately  full  brush  com- 
mence at  the  upper  outline,  and  carry  the  color  along  uniformly  from  left 
to  right  and  from  right  to  left  in  horizontal  bands,  taking  care  not  to  over- 
run the  outlines,  in  approaching  which  the  point  of  the  brush  should  be 
used,  and  at  the  lower  outline  let  there  be  only  sufficient  color  in  the 
brush  to  complete  the  tinting. 

~No  color  should  be  allowed  to  accumulate  in  inequalities  of  the  paper, 
but  should  be  evenly  distributed  over  the  whole  surface. 

Too  much  care  cannot  be  given  to  the  first  application  of  color  ;  as  any 
attempt  to  remedy  a  defect  by  washing  or  applying  fresh  tints  will  be 
found  extremely  difficult,  and  to  generally  make  bad  worse. 

Erasers  should  never  be  used  on  a  tinted  drawing  to  remove  stains  or 


382  TOPOGRAPHICAL   DK AWING. 

patches,  as  the  paper  when  scratched,  receives  the  tint  more  readily,  and  re- 
tains a  larger  portion  of  color  than  other  parts,  thereby  causing  a  darker  tint. 

Marbling  is  done  by  using  two  separate  tints,  and  blending  them  at 
their  edges.  A  separate  brush  is  required  for  each  tint ;  before  the  edge 
of  the  first  is  dry,  pass  the  second  tint  along  the  edge,  blending  one  tint 
into  the  other,  and  continue  with  each  tint  alternately. 

In  reference  to  the  general  effect  to  be  produced  in  tinted  topographi- 
cal drawings,  as  to  intensity,  every  thing  should  be  subordinate  to  clear- 
ness, no  tint  should  be  prominent  or  obtrusive.  Tints  that  are  of  small 
extent  must  be  a  little  more  intense  than  large  surfaces,  or  they  will  appear 
lighter  in  shade.  Keep  a  general  tone  throughout  the  whole  drawing. 
Beginners  will  find  it  best  to  keep  rather  low  in  tone,  strengthening  their 
tints  as  they  acquire  boldness  of  touch. 

In  lettering  tinted  drawings,  let  the  letters  harmonize  with  the  rest  of 
the  plan ;  let  them  be  in  tint  more  intense  than  the  topography,  prominent 
but  not  obtrusive. 

Flourishes  around  the  titles  may  be  used  on  handsome  estate  maps,  and 
on  engraved  maps  of  countries.  They  should  be  used  in  proportion  to  the 
degree  of  finish  bestowed  on  the  rest  of  the  map ;  and  while  they  give 
grace  and  elegance  to  the  title  when  used  in  moderation,  care  should  be 
taken  to  prevent  their  having  too  prominent  an  appearance. 

We  would  recommend  to  every  one  who  wishes  to  make  himself  a 
perfect  draughtsman,  that  he  should  collect  good  charts  and  drawings, 
study  them,  and  in  his  leisure  moments  copy  them.  In  this  way  he  will 
acquire  a  readiness  of  manipulation,  and  ease  and  freedom  of  expression. 

Plate  XCIY.  is  a  map  of  the  Harbor  and  City  of  New  Haven,  re- 
duced from  the  charts  of  the  U.  S.  Coast  Survey,  without  the  depth  of 
soundings  or  the  marks  of  shoals. 

Plates  XCY.  and  XCYI.  are  examples  of  topographical  drawings,  the 
one  in  ink  and  the  other  in  color. 

Plate  XCYII.  is  a  geological  map  from  "  Blake's  Geological  Survey  of 
California."  On  geological  maps  sections  are  similarly  represented,  and 
plans  are  colored  in  patches  according  to  the  formation.  Shades  of  India 
ink  usually  represent  coal  measures ;  of  blue,  limestone ;  of  pink,  the  igne- 
ous rocks,  as  trap,  granite,  &c.  In  all  cases,  there  are  small  blocks  of 
color  at  the  margin  of  the  map,  to  designate  the  mineral  represented  by 
each. 


TOPOOEAPHICAL  DKAWING. 


383 


!•  Private  houses  (occupied  by  persons  not  in  receipt  of  wages). 

2.  Offices  and  shops. 

3.  Houses  occupied  by  persons  in  receipt  of  wages. 

4.  Warehouses. 

5.  Stables  and  outhouses. 

6.  Public  buildings 

7.  Contours,  vertical  distances  between  lines,  two  feet. 

8.  Sewers. 

9.  Gas-pipes. 
10.  Water-pipes. 


The  above  map  is  a  portion  of  the  city  of  London,  taken  from  a  Sanitary 
Report  by  a  Commission  of  Parliament  ;  and  embodies  in  a  graphic  way 
the  details  in  regard  to  drainages,  natural  and  artificial,  contour  lines  and 
street  sewers  ;  position  of  gas  and  water  mains,  and  occupancy  of  buildings. 
On  the  original  are  also  given  the  number  of  the  houses  and  names  of 
streets. 

Reference  has  been  made  to  the  drawing  of  hills  by  contours,  and  it 
has  not  been  recommended  except  when  the  lines  have  been  accurately 


384 


TOPOGRAPHICAL   DRAWIXG. 


determined  by  level.  When  this  is  the  case,  they  should  always  be  used ; 
it  is  the  simplest  and  most  explanatory  record  of  facts,  and  if  the  facts  have 
been  worth  determining  they  are  worth  recording.  When  contour  lines 
are  brought  more  closely  together  (as  shown  in  the  cut  below  from  the 
same  Sanitary  Eeport  and  of  a  larger  portion  of  London),  it  produces  the 
effect  of  physical  relief,  and  shows  at  a  glance  the  lines  of  natural  drainage, 
and  from  it  profiles  can  be  made  in  any  direction,  for  the  grading  of  streets 
or  sewers.  Were  town  and  county  maps  thus  drawn  with  contour  lines, 


much  time  and  money  would  be  saved  in  the  location  of  highways  and 
railways. 

Having  thus  illustrated  within  the  limits  of  our  page  the  different 
kinds  of  topographical  drawing,  we  should  recommend  to  every  one,  who 
wishes  to  become  a  good  draughtsman,  to  collect  good  charts  and  drawings, 
study  and  copy  them,  to  acquire  readiness  of  manipulation  and  ease  and 
freedom  of  expression. 


PERSPECTIVE   DEAWING. 


385 


PEESPECTIYE  DRAWING. 


Fig.  1. 


THE  science  of  Perspective  is  the  representation  by  geometrical  rules,  upon 
a  plane  surface,  of  objects  as  they  appear  to  the  eye,  from  any  point  of 
view. 

All  the  points  of  the  surface  of  a  body 
are  visible  by  means  of  luminous  rays 
proceeding  from  these  points  to  the  eye. 
Thus,  let  the  line  A  B  (fig.  1)  be  placed 
before  the  eye,  C,  the  lines  drawn  from 
the  different  points  1,  2,  3,  4,  &c.,  repre- 
sent the  visual  rays  emanating  from  each 
of  these  points.  It  is  easy  to  understand 
that,  if  in  the  place  of  a  line  a  plane  or 
curved  surface  is  substituted,  the  result  will  be  a  cone  of  rays. 

Let  A  B  (fig.  2)  be 
a  straight  line,  and  let 
the  globe  of  the  eye  be 
represented  by  a  cir- 
cle, and  its  pupil  by 
the  point  C.  The  ray 
emanating  from  A,  en- 
tering through  C,  will 
proceed  to  the  retina 
of  the  eye,  and  be  de- 
picted at  a.  And  as 
it  follows  that  all  the 
points  of  A  B  will  send 
rays,  entering  the  eye  Fig.  2. 

through  0,  the  whole  image  of  AB  will  be  depicted  on  the  retina  of  the 
25 


386  PERSPECTIVE  DRAWING. 

eye  in  a  curved  line  a  3  b.  Conceive  the  line  AB  moved  to  a  greater 
distance  from  the  eye,  and  placed  at  A'  B',  then  the  optic  angle  will  be 
reduced,  and  the  image  a!  3  b'  will  be  less  than  before ;  and  as  our  visual 
sensations  are  in  proportion  to  the  magnitude  of  the  image  painted  on  the 
retina,  it  may  be  concluded  that  the  more  distant  an  object  is  from  the  eye, 
the  smaller  the  angle  under  which  it  is  seen  becomes,  and  consequently 
the  farther  the  same  object  is  removed  from  the  eye  the  less  it  appears. 

Observation  has  rendered  it  evident,  that  the  greatest  angle  under 
which  one  or  more  objects  can  be  distinctly  seen,  is  one  of  90°.  If  be- 
tween the  object  and  the  eye  there  be  interposed  a  transparent  plane  (such 
as  one  of  glass  m  n\  the  intersection  of  this  plane  with  the  visual  rays  are 
termed  perspectives  of  the  points  from  which  the  rays  emanate.  Thus  a 
is  the  perspective  of  A,  b  of  B,  and  so  on  of  all  the  intermediate  points ; 
but,  as  two  points  determine  the  length  of  a  straight  line,  it  follows  that 
a  b  is  the  perspective  of  A  B,  and  a'  b'  the  perspective  of  A''  B'. 

It  is  evident  from  the  figure  that  objects  appear  more  or  less  great  ac- 
cording to  the  angle  under  which  they  are  viewed ;  and  further,  that  ob- 
jects of  unequal  size  may  appear  equal  if  seen  under  the  same  angle. 
For  faswfg,  and  its  perspective  will  be  found  to  be  the  same  as  that  of 
A'B'. 

It  follows  also,  that  a  line  near  the  eye  may  be  viewed  under  an  angle 
much  greater  than  a  line  of  greater  dimensions  but  more  distant,  and 
hence  a  little  object  may  appear  to  be  much  greater  than  a  similar  object 
of  larger  dimensions.  Since,  therefore,  unequally  sized  objects  may  ap- 
pear equal  in  size,  and  equally  sized  objects  unequal,  and  since  objects  are 
not  seen  as  they  are  in  effect,  but  as  they  appear  under  certain  conditions, 
perspective  may  be  defined  to  be  a  science  which  affords  the  means  of  rep- 
resenting, on  any  surface  whatever,  objects  such  as  they  appear  when  seen 
from  a  given  point  of  view.  It  is  divided  into  two  branches,  the  one 
called  linear  perspective,  occupying  itself  with  the  delineation  of  the  con- 
tours of  bodies,  the  other  called  aerial  perspective,  with  the  gradations  of 
colors  produced  by  distance.  It  is  tlie  former  of  these  only,  that  is  pro- 
posed here  to  be  discussed. 

The  perspective  of  objects,  then,  is  obtained  by  the  intersection  of  the 
rays  which  emanate  from  them  to  the  eye,  by  a  plane  or  other  surface 
(which  is  called  the  picture),  situated  between  the  eye  and  the  objects. 

From  the  explanation  and  definition  just  given,  it  is  easy  to  conceive 
that  linear  perspective  is  in  reality  the  problem  of  constructing  the  section, 
by  a  surface  of  some  kind,  of  a  pyramid  of  rays  of  which  the  summit  and 
the  base  are  given.  The  eye  is  the  summit,  the  base  may  be  regarded  as 


PEESPECTIVE   DRAWING. 


387 


the  whole  visible  extent  of  the  object  or  objects  to  be  represented,  and  the 
intersecting  surface  is  the  picture. 

A  good  idea  of  this  will  be  obtained  by  supposing  the  picture  to  be  a 
transparent  plane,  through  which  the  object. may  be  viewed,  and  on  which 
it  may  be  depicted. 

In  addition  to  the  vertical  and  horizontal  planes  with  which  we  are  fa- 
miliar in  the  operations  of  projection,  several  auxiliary  planes  are  em- 
ployed in  perspective,  and  particularly  the  four  following : 


Fig.  3. 

1.  The  horizontal  plane  A  B  (fig.  3),  on  which  the  spectator  and  the 
objects  viewed  are  supposed  to  stand,  for  convenience  supposed  perfectly 
level,  is  termed  the  ground  plane. 

2.  The  plane  M  N",  which  has  been  considered  as  a  transparent  plane 
placed  in  front  of  the  spectator,  on  which  the  objects  are  delineated,  is 
called  the  plane  of  projection  or  the  plane  of  the  picture.    The  intersec- 
tion M  M  of  the  first  and  second  planes  is  called  the  line  of  projection,  the 
ground,  or  base  line  of  the  picture. 

3.  The  plane  E  F  passing  horizontally  through  the  eye  of  the  spectator, 
and  cutting  the  plane  of  the  picture  at  right  angles,  is  called  the  horizontal 
plane,  and  its  intersection  at  D  D  with  the  plane  of  the  picture  is  called  the 
horizon  line,  the  horizon  of  the  picture,  or  simply  the  horizon. 

4.  The  plane  S  T  passing  vertically  through  the  eye  of  the  spectator, 
and  cutting  each  of  the  other  planes  at  a  right  angle,  is  called  the  central 
plane. 

Point  of  view,  or  point  of  sight,  is  the  point  where  the  eye  is  supposed 
to  be  placed  to  view  the  object,  as  at  C,  and  is  the  vertex  of  the 
optic  cone.  Its  projection  on  the  ground  plane  S  is  termed  the  station 
point. 


388 


PEESPECTIVE   DRAWING. 


The  projection  of  any  point  on  the  ground  plane  is  called  the  seat  of 
that  point. 

Centre  of  view  (commonly,  though  erroneously,  called  the  point  of 
sight),  is  the  point  V  where  the  central  vertical  line  intersects  the  horizon 
line  ;  a  line  drawn  from  this  point  to  the  eye  would  be  in  every  way  per- 
pendicular to  the  plane  of  the  picture. 

,  Points  of  distance,  are  points  on  the  horizontal  line,  as  remote  from 
the  centre  of  view  as  the  eye. 

Vanishing  points,  are  points  in  a  picture  to  which  all  lines  converge 
that  in  the  original  object  are  parallel  to  each  other. 

Parallel  perspective. — An  object  is  said  to  be  seen  in  parallel  perspec- 
tive when  one  of  its  sides  is  parallel  to  the  plane  of  the  picture. 

Angular  perspective. — An  object  is  said  to  be  seen  in  angular  perspec- 
tive when  none  of  its  sides  are  parallel  to  the  picture. 

To  find  the  perspective  of  points,  as  the  points  m,  s,  (fig.  4)  in  the  ground 


Fig- 4 

plane,  the  same  letters  designating  similar  planes  and  points  as  in  fig. 
3.  From  the  point  r/i  draw  a  line  to  the  point  of  sight  C,  and  also  to  the 
station  point  S,  at  the  intersection  of  the  line  m  S  with  the  base  line  M  S', 
erect  a  perpendicular  cutting  the  line  in  C,  the  intersection  m1  will  be  the 
perspective  projection  of  the  point  m,  on  the  plane  of  the  picture  M  V. 
The  point  s  being  in  the  central  plane,  its  projection  must  be  in  the  in- 
tersection of  that  plane  by  the  plane  of  the  picture,  as  the  point  s'  the 
intersection  of  the  central  vertical  line  by  the  line  s  C.  The  point  v 
being  both  in  the  central  and  horizontal  plane,  its  projection  in  the  plane 
of  the  picture  must  be  in  the  intersection  of  all  three  planes,  or  at  the 


DRAWING   IN   PERSPECTIVE. 


389 


point  of  view  Y.  The  point  h  being  in  the  horizontal  plane,  its  projec- 
tion must  be  in  the  intersection  of  this  plane  with  the  plane  of  the  picture, 
or  the  intersection  h'  of  the  horizon  line  by  the  line  h  0.  The  points  ti 
and  m  being  in  the  same  vertical  line,  the  points  h'  and  m'  must  also  be  in 
the  same  vertical  line  in  the  plane  of  the  picture,  and  the  position  of  h' 
might  be  determined  by  the  intersection  of  h  C  by  the  perpendicular  to 
the  base  line  at  its  intersection  by  m  S. 

Connect  the  points  hvsm,  and  also  their  projected  perspective  points 
h'  Y  s'  m',  and  we  find  that  when  an  original  line  is  parallel  or  perpendic- 
ular to  the  lase  of  the  picture,  the  perspective  of  that  line  will  also  be  par- 
allel  or  perpendicular  to  it. 

Fig  5.  Draw  the  diagonals  Ms  and  m  S',  project  as  in  preceding  fig- 
ure the  points  m  and  s  into  the  plane  of  the  picture,  draw  M  m,  M  S'  and 
S'  m' ;  now  since  m  and  M  are  the  extremities  of  a  line  perpendicular  to 


Fig.  5. 

the  plane  of  the  picture,  the  line  m'  M  must  be  the  projection  of  this  line 
on  the  plane  of  the  picture,  and  if  this  line  be  extended  it  will  pass  through 
Y,  which  may  be  demonstrated  of  all  lines  perpendicular  to  the  plane  of 
the  picture  ;  hence  the  perspective  direction  of  lines  perpendicular  to  the 
picture  is  to  the  centre  of  view. 

If  the  line  ra'  S'  be  extended,  it  will  pass  through  the  point  D,  and  if 
M  s'  be  extended  it  will  pass  through  a  point  in  the  line  of  the  horizon  at 
a  distance  from  Y  equal  to  Y  D ;  by  construction  D  Y  has  been  made  equal 
to  Y  C,  and  as  this  demonstration  is  applicable  to  other  similar  lines,  and 
since  MmsS'  is  a  square;  hence  the  perspective  direction  of  all  lines, 


390  DRAWING  IN  PERSPECTIVE. 

making  an  angle  of  45°  with  the  plane  of  the  picture,  is  towards  the  point 
of  distance. 

Having  thus  illustrated  the  rules  of  parallel  perspective,  we  now  pro- 
ceed to  apply  them  to  the  drawing  of  a  square  and  cube,  PI.  XCIX.  The 
same  letters  are  employed  in  similar  position  as  in  preceding  figures. 

It  is  necessary  to  premise  that  the  student  should  draw  these  examples 
at  least  three  times  the  size  of  those  in  the  plate. 

Let  A  and  B  (fig  1,)  represent  the  plan,  or  situation  upon  the  ground, 
of  two  squares,  of  which  a  perspective  representation  is  required.  First 
draw  the  line  M  M,  which  represents  the  base  line  of  the  picture ;  make  S 
the  station  point  or  place  of  the  observer,  and  draw  lines  or  rays  from  all 
visible  angles  of  the  squares,  to  S  ;  then  draw  the  lines  S  M,  parallel  to  the 
diagonal  lines  of  the  squares.  Now  draw  M'  M'  parallel  to  M  represent- 
ing the  base  line  of  the  picture  in  elevation ;  then  draw  S'  Y,  the  vertical 
line  immediately  opposite  the  eye ;  let  the  distance,  S'  Y,  be  the  height  of 
the  eye  from  the  ground,  and  draw  D  D  the  horizontal  line ;  Y  being  the 
centre  of  view ;  let  fall  perpendicular  lines  from  the  angles  a  and  J  of  the 
plan  of  the  square  A,  and  also  from  the  point  c,  where  the  ray  from  the 
angle  e  intersects  the  base  line,  M  M,  and  from  a'  and  5',  where  a  a'  and  5  V 
intersect  the  base  or  ground  line  M'  M',  draw  lines  to  the  centre  of  view, 
Y;  and  e'  where  the  perpendicular  line  from  c  intersects  the  line  5' Y,  will 
give  the  apparent  or  perspective  width  of  the  side  b  e ;  from  e'  draw  a 
line  parallel  to  a'  5',  and  the  perspective  representation  of  the  near- 
est square  A,  is  complete.  In  order  to  prove  the  accuracy  of  this  per- 
formance, it  is  necessary  to  try  if  the  diagonal  lines,  a'  ef,  and  V  f,  incline 
respectively  to  the  points  of  distance,  D  D,  on  the  horizontal  line :  if  so,  it 
is  correct.  The  square  B  is  drawn  in  precisely  the  same  manner,  and  will 
be  easily  understood  by  observing  the  example. 

The  plans  of  the  two  cubes  C  and  D,  are  the  same  as  the  plans  of  the 
squares  A  and  B.  As  neither  of  these  cubes  appears  to  touch  the  plane 
of  the  picture  M  M,  it  will  be  necessary  to  imagine  the  sides  I  </,  and  ~k  h, 
to  be  continued  until  they  do  so ;  now  draw  down  perpendicular  lines  from 
where  the  continuations  of  these  sides  intersect  the  base  line,  and  set  off 
on  them  from  the  line  M'  M',  the  height  of  the  cube,  as  1 — 2  which 
is  the  same  as  the  width,  and  complete  the  square  shown  by  the  dotted 
lines :  from  all  four  angles  of  this  square  draw  lines  to  the  centre  of  view 
— this  will  give  the  representation  of  four  lines  at  right  angles  with  the 
picture  carried  on  as  far  as  it  would  be  possible  to  see  them ;  then  it  only 
remains  to  cut  off  the  required  perspective  widths  of  the  cubes  by  the  per- 
pendicular lines  from  the  intersection  of  the  visual  rays  with  the  plane  of 


PERSPECTIVE  DRAWING.  391 

the  picture :    the  completion  of  this  problem  will  be  very  easy,  if  the 
drawing  of  the  squares  is  well  understood. 

In  such  simple  objects  as  these  it  will  not  be  necessary  to  draw  a  plan  ; 
when  one  side  is  parallel  to  the  picture,  and  dimensions  are  known.  In 
fig.  2,  the  same  objects  as  those  in  fig.  1  are  drawn  without  a  plan  thus  :— 

Draw  the  ground  line  M  M,  then  the  vertical  line  S'  Y,  and  the  horizon- 
tal line  D  D,  at  the  height  of  the  eye ;  making  D  D  the  same  distance  on 
each  side  of  V,  that  the  eye  is  from  the  transparent  plane;  for  drawing 
the  squares  mark  off  from  S'  to  &',  on  the  ground  line,  the  distance  that 
the  square  is  on  one  side  of  the  observer ;  let  V  a'  be  the  length  of  one 
side  of  the  square ;  from  V  and  a'  draw  lines  to  Y,  which  represent  the 
sides  of  the  square  carried  on  indefinitely ;  to  cut  off  the  required  per- 
spective width  of  the  side  I'  e'  of  the  square,  lay  off  the  width,  a'  V,  from 
b'  top,  then  draw  from^>  to  D  on  the  left,  and  the  point  e'  where  the  line 
Dp  intersects  V  Y,  will  give  the  apparent  width  required ;  then  draw/'  e' 
parallel  to  a'  &',  and  the  square  is  complete :  this  may  be  proved  in  the 
same  way  as  in  fig.  1 .  The  further  square  may  be  obtained  in  a  similar 
manner,  setting  off  the  distance  between  the  squares  from^  to  £,  and  the 
width  of  the  square  beyond  that,  and  drawing  lines  to  D  as  before: 
some  of  the  lines  in  this  plate  are  not  continued  to  the  ground  line, 
in  order  to  avoid  confusion.  Proceed  with  the  cubes  by  the  same  rule. 
Let  1,  2,  3,  4,  be  the  size  of  one  side  of  the  cube  if  continued  until  touch- 
ing the  picture  ;  from  these  points  draw  rays  to  Y :  from  3  to  t  set  off  the 
distance  the  cube  is  from  the  picture,  and  from  t  to  r,  the  width  of  the 
cube  ;  draw  from  these  points  to  D  on  the  right,  and  their  intersection  of 
the  line  3  Y  in  m,  <?,  will  give  the  perspective  width  and  position  of  that 
side  of  the  cube :  draw  lines  perpendicular  to  the  ground  line  from  m  and 
e>,  and  lines  parallel  to  4 — 2  from  the  angles  of  the  cube,  l',gf,m;  then 
draw  the  side  n  A',  and  the  cube  is  complete.  The  operation  of  drawing 
the  other  cube  is  similar,  and  easy  to  be  understood. 

From  the  drawing  of  a  square  in  parallel  perspective,  we  deduce  rules 
for  the  construction  of  a  scale  in  perspective.  Let  D  M  M  D,  (fig  6,)  be 
the  plane  of  the  picture,  the  same  letters  of  reference  being  used  as  in 
preceding  figures.  From  S'  lay  off  the  distance  o  S'  equal  to  some  unit 
of  measure,  as  may/be  most  convenient ;  from  o  draw  the  diagonal  to  D 
the  point  of  distance ;  now  draw  1  V  parallel  to  the  ground  line  M  M, 
again  draw  from  I/  the  diagonal  V  D,  and  lay  off  the  parallel  2  2',  pro- 
ceed in  the  same  way  with  the  diagonal  2'  D  and  the  parallel  3  3',  and 
extend  the  construction  as  far  as  may  be  necessary.  It  is  evident  o  S'  1 1/, 
V  1  2  2',  2'  2  3  3'  are  the  perspective  projectors  of  equal  squares,  and 


392 


PEESPECTIVE   DRAWING. 


therefore  o  S',  1 1',  2  2'  3  3',  etc.,  and  S'  1,  1  2,  2  3,  etc.,  are  equal  to  each 
other,  and  that  if  o  S'  is  set  off  to  represent  any  unit  of  measure,  as  one 
foot,  one  yard,  or  ten  feet,  &c.,  each  of  these  lines  represents  the  same  dis- 


JIT 


Fig.  6. 


tance,  the  one  being  measures  parallel  to  the  base  line,  the  others  perpen- 
dicular to  it.  In  making  a  perspective  drawing  a  scale  thus  drawn  will  be 
found  very  convenient ;  but  as  in  the  centre  of  the  picture  it  might  inter- 
fere with  the  construction  lines  of  the  object  to  be  put  in  perspective,  it  is 
better  that  the  scale  be  transferred  to  the  side  of  the  picture  a  M  o,  the  di- 
agonals to  be  laid  off  to  a  point  to  the  right  of  D  equal  to  the  point  of 
distance. 

The  scales  thus  projected  are  for  lines  in  the  base  or  ground  plane ;  for 
lines  perpendicular  to  this  plane  the  following  construction  is  to  be  adopt- 
ed ;  upon  any  point  of  the  base  line  removed  from  S',  as  a  for  instance, 
erect  a  perpendicular,  ad /  on  this  line,  lay  off  as  many  of  the  units  o  S' 
as  may  be  necessary ;  in  this  example  three  have  been  laid  off,  that  is,  a  d 
=3  o  S'.  From  a  and  d  draw  lines  to  the  centre  of  view,  and  extend  the 
parallels  1 1',  2  2',  3  3  ' ;  at  the  intersection  of  these  lines  with  a  Y  erect 
perpendiculars.  The  portions  comprehended  between  the  lines  a  Y  and  d 
Y  will  be  the  perspective  representations  of  the  line  a  d,  in  planes  at  dis- 
tances of  1,  2,  3,  o  S'  from  the  base  line,  and  as  5,  c,  d  are  laid  off  at  inter- 
vals equal  to  o  S',  by  drawing  the  lines  c  Y  and  5  Y  six  equal  squares  are 
constructed,  of  which  the  sides  correspond  to  the  unit  of  measure,  0  S'. 

To  determine  the  Perspective  Position  of  any  point  in  the  Ground  Plane. 
Thus  (fig.  7),  to  determine  the  position  of  the  point  p,  which  in  plane 
would  be  six  feet  distant  from  the  plane  of  the  picture,  M  M,  and  ten  feet 
from  the  central  plane,  to  the  left. 

Lay  off  from  S',  to  the  left,  the  distance  a  S',  equal  to  six  feet  on  the 


PERSPECTIVE   DRAWING. 


393 


scale  adopted ;  draw  the  diagonal  to  the  point  of  distance  D,  on  the  right ; 
at  its  intersection  a!  with  the  vertical  line  V  S',  draw  a  parallel  to  M  M 
the  base  line;  lay  off  from  S',  S'  ~b  equal  to  ten  feet,  draw  5  V;  the  inter- 
section of  this  line  p,  with  the  parallel  previously  drawn,  will  be  the  posi- 
tion of  the  point  required. 


Fig.  7. 

By  a  similar  construction  the  position  of  any  point  in  the  ground  plan 
may  be  determined.  It  is  not  necessary  that  the  distances  should  be  ex- 
pressed numerically ;  they  may  be  shown  on  the  plan  and  thence  be  trans- 
ferred to  the  base  line,  and  thrown  into  perspective  by  the  diagonals  and 
parallels.  As  the  intersections  of  the  various  lines  of  the  outlines  of  ob- 
jects are  points,  by  projecting  perspectively  these  points,  and  afterwards 
connecting  by  lines,  the  perspective  of  any  plane  surface,  on  the  ground 
plane,  may  be  shown. 

If  the  point  p  were  not  in  the  ground  plane,  but  in  a  position  directly 
above  that  already  assumed,  that  is,  the  distances  from  the  plane  of  the  pic- 
ture, and  the  central  plane  being  the  same,  but  its  distances  above  the 
ground  plane  were,  say,  five  feet ;  then  at  5  erect  a  perpendicular,  and  lay 
off  b  V  equal  to  five  feet,  connect  V  Y,  at  p  erect  another  perpendicular, 
and  its  intersection  p'  with  the  line  V  Y  will  be  the  position  of  the  point 
required. 

Or  the  plane  of  the  point  p1  might  be  assumed  as  the  position  of  the 
ground  plane,  M'  M'  becoming  the  base  line,  and  laying  off  from  S",  S"  a" 
and  S"bf — equal  respectively  to  six  and  ten  feet;  drawing  the  diagonal 
a"D  and  V  Y  and  the  parallel  as  before,  the  point  p'  will  be  determined. 

To  draw  an  Octagon  in  Parallel  Perspective, — Let  A  (fig.  8)  represent 
the  plan  of  an  octagon.  Draw  M  M,  S'  Y,  and  D  D,  as  before  ;  from  the 
points  M,  a,  J,  0,  draw  rays  to  Y.  Set  off  on  M  M  from  c  to  the  right  the 
distances  ce,  cd,  cf,  from  which  draw  diagonals  to  D  on  the  left,  and 
at  their  intersection  with  the  ray  c  Y,  draw  parallels  e'  g',  d'  A',  ~k'  I',  to 
the  base  line ;  these  points  will  correspond  to  the  angles  on  the  plan.  Now 


394: 


PECTIVE   DRAWING. 


connect  the  angles  on  the  perspective  view,  in  the  proper  succession,  and 
the  perspective  projection  is  complete. 

It  will  be  observed,  that  in  this  construction  the  plan  has  been  placed 
forward  of  the  plane  of  the  picture,  contrary  to  the  position  it  should  oc- 
cupy, which  should  be  the  same  relative  position  back  of  this  plane ;  but 
it  will  be  found  much  simpler  in  construction  than  if  it  were  placed  as  in 
PI-  XCIX.  and  the  points  were  all  projected  to  the  base  line ;  it  is,  of  course, 
equally  correct  in  its  perspective  projection. 


D 


Fig.  8. 

To  draw  a  Circle  in  Parallel  Perspective. — Let  C,  (fig.  8)  represent  the 
plan  of  a  circle,  round  which  let  the  square  a  e  c  m  be  described,  two  of 
its  sides  being  parallel  to  the  base  line  M  M ;  draw  diagonals  across  the 
square,  and  where  these  intersect  the  circumference  of  the  circle  draw  the 
lines  bk  and  dg  parallel  to  the  base  line,  and  the  lines  o  n  and^?^  at  right 
angles  thereto.  Draw  also  the  lines  f  I  and  c  A  at  right  angles  to  each 
other  through  the  centre  of  the  circle,  project  the  points  #,  0,  l,p,  m,  to 
the  base  and  draw  rays  to  Y ;  set  off  from  a'  to  the  left  the  distances  a' «, 
«7>,  a'c,  a'd,  a'e,  and  draw  diagonals  to  the  point  of  distance  D  on  the 
right ;  at  their  intersection  with  the  line  a'  V  draw  horizontal  lines,  or 
parallels  to  the  base,  and  there  will  be  projected  in  perspective  the  square 
aecm,  with  all  the  lines  of  parallels  and  perpendiculars ;  connect  the 
intersections  corresponding  to  the  points  c,  n,  f,  g,  h,  k,  Z,  r,  and  we 


PERSPECTIVE  DRAWING. 


395 


have  the  perspective  projection  of  the  required  circle,  which  will  be  an 
ellipse. 

To  erect  upon  the  octagonal  base  A  an  octagonal  pillar  or  tower.  This 
construction  resolves  itself  into  simply  constructing  another  octagon  on  an 
upper  plane,  and  connecting  the  visible  angles  by  perpendiculars,  or  per- 
pendiculars may  be  erected  at  the  points  M,  #,  J,  c,  and  the  heights  of  the 
tower  laid  off  upon  them,  and  from  these  extremities  rays  drawn  to  the 
centre  of  view ;  the  intersection  of  these  rays  by  perpendiculars  from  the 
angles  of  the  octagon  beneath  will  determine  the  projection  of  the  upper 
surface  of  the  pillar;  represent  in  full  lines  all  visible  outlines,  and  the 
projection  is  complete. 

In  the  same  manner  a  pillar  may  be  erected  on  the  circular  base.  If 
the  pillars  be  inclined,  the  first  method  of  projecting  the  upper  outline  on 
a  plane  assumed  at  the  height  of  the  pillar,  must  be  adopted. 

To  draw  a  Pyramid  in  Parallel  Perspective. — Let  A  (fig.  9)  be  the 
plan  of  a  pyramid,  the  diagonal  lines  represent  the  angles,  and  their  in- 
tersection the  vertex ;  project  the  plan  as  in  previous  examples  of  squares. 
Draw  diagonal  lines  from  M  to  5,  and  a  to  c,  their  intersection  gives  the 
perspective  centre  of  the  square ;  upon  this  point  raise  a  perpendicular 
line  which  is  the  axis  of  the  pyramid ;  draw  a  perpendicular  line  e  /,  in 


D 


Fig.  9. 

the  centre  of  the  line  M  a,  upon  which  set  up  the  height  of  the  pyramid 
ef;  from/  draw  a  line  to  V,  and  its  intersection  of  the  axis  of  the  pyramid 
at  d  will  give  the  perspective  height ;  complete  the  figure  by  drawing 
lines  from  d,  the  apex,  to  M,  a,  5,  the  three  visible  angles.  The  other 
two  pyramids  are  drawn  in  a  similar  manner,  by  setting  their  distances 
from  the  plane  of  the  picture  off  from  a,  on  the  ground  line  to  the  right, 
and  drawing  diagonals  to  the  point  of  distance  on  the  left. 


396  PERSPECTIVE 

To  draw  a  Cone  in  Parallel  Perspective. — Let  B  (fig.  9)  represent  the 
plan  of  a  cone,  apply  the  same  lines  of  construction  as  to  C  (fig.  8) ;  and 
draw  the  perspective  view  of  a  circle,  upon  the  perspective  centre  of 
which  draw  a  perpendicular  line,  a,  J>  /  on  the  centre  of  the  line  d  <?,  raise 
a  perpendicular,  upon  which  set  up  the  height  of  the  cone,  from  the 
ground  line  to  c  j  from  c  draw  a  ray  to  Y,  and  the  point  where  this  line 
intersects  the  axis  of  the  cone  a  &,  in  &,  will  give  the  perspective  height 
of  the  axis ;  from  5  draw  lines  toy  and  g,  and  the  figure  is  complete. 

To  draw  the  prism,  C,  which  consists  of  two  triangular  ends  and  three 
rectangular  sides,  place  the  length  of  the  side  a  M  upon  the  ground  line, 
and  draw  lines  to  V ;  mark  off  the  width  of  one  end  from  a  to  the  left  upon 
the  ground  line,  and  jd¥aw  to  the  point  of  distance  on  the  right,  which 
gives  the  perspective  width,  ad;  find  the  perspective  centre  f  of  the 
side  in  the  same  way,  and  from  f  and  d  draw  horizontal  lines  until  they 
intersect  the  line  from  M  Y ;  upon  f  and  g  draw  perpendiculars ;  set  up 
the  height  a  5,  of  the  end  of  the  prism,  and  from  b  draw  a  line  lo  Y,  and 
the  point  where^  it  intersects  f  c,  in  c,  will  give  the  perspective  height  of 
the  end  of  the  figure  ;  from  G  draw  c  e,  parallel  to  a  M,  from  c  draw  c  d, 
and  c  a,  and  the  visible  end  is  complete ;  the  other  end  is  dotted  in  to  show 
the  process  only. 

E  is  the  representation  of  a  cylinder,  with  one  end  towards  the  spec- 
tator; its  projection  will  be  easily  understood  by  examination. 

To  draw  a  Square  and  Cube  in  Angular  Perspective.  Plate  C.  Let 
A  (fig.  1)  be  the  plan  of  the  square,  and  B  the  plan  of  the  cube,  M  M  the 
base  or  ground  line,  and  S  the  station  point.  Draw  M'  M',  and  D  D'  par- 
allel to  M  M,  the  one  being  the  ground  line  and  the  other  the  horizon  of 
the  plane  of  the  picture ;  project  the  point  d  on  M  M  to  d,  on  M'  M'.  It 
has  been  shown  in  parallel  perspective  that  the  vanishing  points  of  diago- 
nals of  squares  lie  in  the  points  of  distance ;  if  through  the  station  point 
S,  in  any  of  the  preceding  figures,  lines  be  drawn  parallel  to  the  diagonals, 
they  will  intersect  the  base  lines  at  distances  from  the  'central  plane  equal 
to  the  points  of  distance.  In  like  manner  to  find  the  vanishing  points  of 
lines  in  the  ground  planes,  or  in  planes  parallel  to  the  ground  plane,  in- 
clined to  the  plane  of  the  picture,  through  the  station  point  S  draw  lines  par- 
allel to  the  inclined  lines,  and  project  their  intersection  with  the  base  line 
to  the  horizon  of  the  picture ;  thus,  in  the  present  example  draw  S  M,  S  M 
parallel  to  ad,  e  A,  and  to  dc,  kg ;  project  their  intersections  M,  M,  with 
the  base  line  to  D,  D',  the  horizon  of  the  picture,  and  D,  D',  will  be  the 
vanishing  points  of  all  lines  parallel  to  a  d  and  d  c.  Draw  d'  D  and  d'  D', 
the  perspective  projection  of  d  a  will  lie  in  the  former  of  these  lines  and 


PERSPECTIVE   DKAWING.  397 

d  c  in  the  latter.  To  determine  the  perspective  position  of  the  points  a 
and  c,  or  the  length  of  these  lines,  draw  the  rays  a  S  and  <?S,  project  their 
intersection  with  the  base  M  H,  upon  the  lines  'd1  D  and  d'  D',  and  their  in- 
tersections a',  c'  will  be  the  perspective  projection  of  the  points  a  and'c. 
To  complete  the  projection  of  the  square,  draw  the  lines  a!  D'  and  c  D, 
their  intersection  will  be  the  perspective  projection  of  the  point  5,  and  the 
square  is  complete.  To  prove  the  construction,  draw  the  ray  I  S  and  pro- 
ject its  intersection  with  the  base  M  M,  and  if  the  construction  be  correct 
it  will  fall  upon  the  point  V . 

As-  the  cube  is  placed  at  some  distance  from  the  plane  of  the  picture, 
it  will  be  necessary  to  continue  either  e  h  or  g  h,  or  both,  till  they  intersect 
the  base  line  M  M  at  n  and  m;  drop  perpendiculars  or  project  these  points 
upon  M  M'  at  n'  and  m' ;  on  these  perpendiculars  set  up  the  height  of  the 
cube  m!  o  and  n'  s,  draw  the  lines  in'  D',  o'  ~D'  and  n'  D,  s  D ;  connect  the 
intersections  Ji  and  h"  ;  draw  the  rays  S  e  and  S  #-,  and  project  their  inter- 
sections with  M  M,  to  g'  e' ;  draw  the  lines  e"  D'  and  g"  D ;  if  the  construction 
be  correct,  the  projection  of  the  intersection  of  the  ray  S/with  the  base 
will  fall  upon/',  and  of  the  ray  S  h  will  fall  upon  h"  and  ti. 

To  Solve  the  Same  Problem  ly  a  Different  Construction. — Let  A  and 
B,  (fig.  1,)  be  as  before  the  plans  of  the  square  and  of  the  cube ;  to  pro- 
ject them  perspectively  on  the  plane  of  the  picture  MD  D'M,  (fig.  2). 

From  the  point  M  and  M,  (fig.  1,)  set  off  distances  equal  to  M  S,  M  S, 
to^>  and^/  ;  project  these  points  upon  D  D'  fig.  2,  the  pointy,  fig  2,  will  be 
that  from  which  any  number  of  parts  may  be  laid  off  on  lines  vanishing  in 
D' ;  the  pointy  will  be  the  corresponding  point  for  lines  vanishing  in  D. 
These  points  may  be  called  the  points  of  division.  In  parallel  perspec- 
tive the  points  of  distance  were  the  points  of  division,  the  one  for  the 
other.  To  illustrate  their  application  in  the  present  example,  project  the 
point  d,  (fig.  1,)  to  d'  (fig.  2,)  draw  d'  D  and  d'  D',  from  d'  on  either  lay 
off  a  distance  d'  i,  d'  Tt,  equal  to  the  side  of  the  square  a  d.  Now  since  p 
is  the  division  point  of  lines  vanishing  in  D  from  *',  draw  the  line  ip,  and 
its  intersection  with  d'  D  cuts  off  a  line  d'  o!  equal  perspectively  to  the 
line  d'  i  or  ad  measured  on  the  base  line.  Again  since p'  is  the  division 
point  of  lines  vanishing  in  D',  the  line  fop'  cuts  off  on  d'  D',  a  line  d'  c'  equal 
perspectively  to  the  line  d'  &,  or  a  d  measured  on  the  base :  having  a'  d'  c, 
the  square  is  completed  by  drawing  the  lines  c'  V  towards  D,  and  a'  b' 
towards  D'. 

To  construct  the  cube,  project  the  point  m,  (fig.  1,)  to  m',  (fig.  2) ;  lay 
off  on  the  perpendicular  forming  the  projection,  the  height  m'  o'  of  the 
cube ;  draw  the  lines  m'  D'  and  o'  D'.  Lay  off  the  disfcnce  m!  r  equal  to 


398  PERSPECTIVE   DRAWING. 

m  h,  (fig.  1,)  and  draw  the  line  rpf,  its  intersection  with  m'  D'  will  cut  off 
m!  Jif)  equal  to  m  h,  (fig.  1,)  and  establish  the  angle  h  of  the  cube.  From 
r  lay  o&rs,  equal  to  hg,  (fig.  1,)  draw  sp1,  and  its  intersection  with  m!  D' 
establishes  the  angle  g1 '.  From  h'  draw  a  line  vanishing  in  D.  Through. 
A'  extend  a  line  p  h'  to  £,  from  t  lay  off  to  the  left  t  a,  equal  to  the  side  of 
the  cube  he;  draw  ap,  and  its  intersection  with  the  line  h'  D,  establishes 
a  third  point  e  of  the  cube.  Upon  these  points  h'  (f  e'  erect  perpendicu- 
lars ;  those  upon  h'  and  g'  will,  by  their  intersection  with  o'  D,  determine 
h"  g".  Draw  h"  D',  its  intersection  with  the  perpendicular  at  G  determines 
e".  Draw  g"  D'  and  e"  D  to  their  intersection,  and  the  cube  is  complete. 

To  Draw  the  Perspective  Projection  of  an  Octagonal  Pillar  in  Angu- 
lar Perspective. — Plate  GI.  Let  A,  (fig.  1,)  be  the  plan  of  the  pillar.  En- 
close it  by  a  square.  Let  M  M  be  the  base  line,  and  S  the  station  point ;  de- 
termine the  position  of  the  vanishing  points  for  the  sides  of  the  square  as  in 
Plate  0.,  and  project  the  square  upon  the  plane  of  the  picture  M  D  D'  M' 
by  either  of  the  methods  already  explained.  These  lines  of  construction  are 
omitted,  as  on  the  necessarily  small  diagrams  they  would  confuse  the  stu- 
dent ;  but  in  drawing  these  examples  to  the  scale  recommended,  they  might 
be  retained.  From  the  angles  of  the  octagon  visible  to  the  spectator 
draw  rays  to  the  station  point  S  project  their  intersection  with  the  base 
line  MM,  to  the  perspective  square,  (fig.  2,)  which  will  thus  determine  on 
the  sides  of  the  square  the  positions  of  the  points  #',  5',  <?',  d',  e',  correspond- 
ing to  the  visible  angles  of  the  octagon  ;  connect  these  points  by  lines.  To 
construct  the  pillar  upon  this  base,  upon  the  perpendicular  let  fall  from 
the  corner/"  of  the  square  upon  M  M'  at/"  set  off  the  height  of  the  pillar ; 
from  this  pointy  draw  lines  to  the  vanishing  points  D,  D',  and  construct 
three  sides  of  an  upper  square  similar  to  the  lower  one.  The  lines  of  this 
square  will  determine  the  length  of  the  sides  of  the  tower,  which  are  the 
perpendiculars  10t  fall  upon  a1  V  c'  d'  e'. 

To  Construct  a  Circular  Pillar  in  Angular  Perspective. — Plate  CI. 
Let  B,  (fig.  1,)  be  the  plan  of  the  base ;  enclose  it  with  a  square  whose  sides 
are  parallel  respectively  to  S  MandS  M;  project  this  square  upon  the  plane 
of  the  picture,  (fig.  2,)  divide  the  plan  into  four  equal  squares  by  lines  par- 
allel to  the  sides  ;  draw  rays  through  the  points  h  and  i,  and  project  their 
intersection  with  M  M  upon  the  perspective  square.  From  the  points  hf  and 
i'  thus  formed,  draw  lines  to  vanishing  points  D'  and  D,  and  the  perspec- 
tive square  is  divided  similarly  to  the  original,  and  there  are  four  points  of 
the  circle  established :  through  these  draw  the  perspective  of  the  circle. 
By  the  division  of  the  base  into  smaller  squares  more  points  of  the  curve 
might  be  determined,  but  for  the  present  purpose  they  are  unnecessary. 


PERSPECTIVE  DRAWING.  399 

To  determine  the  outline  of  the  pillar,  draw  from  S  rays  tangent  to  the 
sides  of  the  plan  at  Jc  and  *,  the  perpendiculars  let  fall  from  their  intersec- 
tion with  M  M  will  be  the  outline  of  the  cylinder.  To  cut  them  off  to  the 
proper  height,  and  to  determine  the  top  of  the  cylinder,  upon  the  perpen- 
dicular let  fall  upon  *',  set  off  the  height  of  the  cylinder  I'  I",  and  upon  this 
plane  project  the  square  as  before,  and  draw  in  through  the  points  thus 
determined  the  outline  of  the  curve.  As  a  still  further  elucidation  of  the 
principle  of  projection,  an  enlarged  cap  is  represented  on  the  pillar,  of 
which  the  circumscribing  circle  (fig.  1,)  is  the  plan.  In  this  by  extending 
the  central  lines  of  the  square,  both  in  plan  and  perspective,  we  are  en- 
abled to  project  readily  eight  points  in  the  larger  circle  through  which 
the  curve  may  be  drawn. 

To  Draw  an  Octagonal  Pyramid  in  Angular  Perspective. — Plate  CI. 
Let/",  (fig.  1,)  be  the  base  of  the  pyramid;  project  upon  the  plane  of  the 
picture,  (fig.  3,)  the  visible  angles  of  the  base,  as  in  the  case  of  the  pillar. 
Through  the  centre  of  the  plan  draw  a  line  parallel  to  one  of  the  sides  and 
intersecting  MM  at  m /  from  this  point  let  fall  a  perpendicular  to  mf  on 
M  Mx,  (fig.  3,) ;  on  this  perpendicular  set  off  the  height  of  the  pyramid  m' 
o  from  m!  and  draw  lines  to  D'.  From  the  centre  of  the  plan  draw  a  ray 
to  S,  and  project  its  intersection  with  M  M,  upon  the  line  o  D',  its  intersec- 
tion o'  with  this  line  will  be  the  apex  of  the  pyramid :  from  this  point 
draw  lines  to  the  angles  of  the  base  already  projected,  and  the  pyramid  is 
complete. 

To  Draw  a  Cone  in  Angular  Perspective. — Plate  CI.  Let  the  inner 
circle  B,  (fig.  1,)  be  the  base  of  the  cone  project  its  visible  outline  to  fig. 
3,  as  in  case  of  the  cylinder.  To  determine  its  height  extend  one  of  the 
diameters  of  the  plan  to  the  base  line  atjp;  from  this  point  let  fall  a  per- 
pendicular to p'  on  MM',  and  set  off  upon  \tp' g,  the  height  of  the  cone ; 
fromy  and  g  draw  lines  to  the*  vanishing  point  D'.  From  the  centre  of 
the  plan,  (fig.  1,)  draw  rays  to  S,  and  project  its  intersection  with  MM, 
upon  r'  on  the  line  g  D',  and  /  will  be  the  apex  of  the  cone :  connect  the 
apex  with  the  extremities  of  the  perspective  of  the  base, and  the  projec- 
tion of  the  cone  is  complete. 

To  Draw  the  Elevation  of  a  Building  in  Angular  Perspective. — Plate 
Oil.  For.  example,  take  the  school-house,  PI.  LXXIY.  of  architecture. 
Plot  so  much  of  the  plan  of  the  building  at  it  as  may  be  seen  from  the  po- 
sition of  the  spectator  at  S.  Draw  a  base  line,  and  through  the  station 
point  draw  parallels  to  the  sides  of  the  building  cutting  the  base  as  at  M 
M :  draw  MM'  for  a  base,  and  DD'  for  the  horizontal  line  of  the  picture. 
Project  M  and  M  to  D  and  D',  for  the  vanishing  points,  the  one  of  the 


400  PERSPECTIVE   DRAWING. 

lines  parallel  to  a  c,  the  other  to  a  I ;  extend  a  c,  a  I ;  project  d,  <?,  to 
d,  e't  and  on  df  d  set  off  the  height  of  the  eaves  d'  o,f  and  of  the  ridge  d 
n ;  from  d',  o  and  n  draw  lines  to  D',  and  from  e  to  D,  draw  rays  from  c 
and  l>  to  S',  and  project  their  intersection  with  the  base  to  the  vanishing 
lines  just  drawn.  To  find  the  perspective  of  the  ridge  draw  a  ray  from 
the  centre  of  a  &,  and  project  its  intersection  with  the  base  to  r  on  the  line 
n  D',  the  point  is  the  apex  of  the  gable,  the  line  r  D  will  be  the  perspective 
of  the  ridge ;  to  determine  its  length  erect  a  perpendicular  at  the  intersec- 
tion of  t  D'  and  s  D,  draw  the  sloping  lines  of  the  roof,  and  the  outline  of 
the  building  is  complete.  The  filling  in  of  the  details  will  be  readily  under- 
stood ;  it  will  only  be  necessary  to  keep  in  mind,  that  all  lines  parallel  to 
a  1)  must  meet  in  D',  those  to  a  c  in  D :  all  measures  laid  off  on  any  lines 
of  the  plan  must  be  connected  with  the  point  of  sight  S,  and  their  inter- 
sections with  the  base  projected.  All  vertical  heights  must  be  laid  off  on 
the  line  d'  d,  and  referred  to  the  proper  position  by  lines  to  D  or  D',  as  the 
case  may  be. 

As  an  example  of  the  other  method  of  constructing  this  same  problem, 
let  the  scholar  lay  off  to  the  double  of  the  present  scale  the  plane  of  the 
picture  M  D  D'  IF,  and  the  division  points  p'  and  p,  and  without  drawing 
plan  or  elevation  take  the  dimensions  from  Plate  LXXIY. 

To  Draw  an  Arched  Bridge  in  Angular  Perspective. — PL  CIII.  Let 
A  and  B,  (fig.  1,)  be  the  plans  of  the  piers;  on  the  line  a  A,  one  of  the  sides 
of  the  bridge,  lay  down  the  curve  of  the  arch  as  it  would  appear  in  eleva- 
tion, in  this  example  an  ellipse.  Divide  the  width  of  the  arch  as  at  b.  c. 
d.  e.f.  g.  A.,  carry  up  lines  perpendicular  to  5  h  until  they  intersect  the 
curve  of  the  arch,  and  through  these  points,  draw  lines  parallel  to  b  h  as  Jc. 
I.  m.  /  let  o  r  be  the  height  of  the  parapet  of  the  bridge  above  the  spring  of 
the  arch.  Through  the  station  point  draw  lines  parallel  to  the  side  a  h 
and  end  a  a  of  the  bridge,  till  they  intersect  the  assumed  base  line  M  M : 
project  these  intersections  to  the  horizon  line  of  the  picture  for  the  vanish- 
ing points  D,  D'  of  perspective  lines  parallel  to  a  h  and  a  a.  Let  fall  a 
perpendicular  from  a  to  a',  and  on  this  perpendicular  set  off  from  a!  the 
heights  s  k,  s  I,  s  m,  and  s  t ;  from  a!  and  r'  draw  lines  to  D  and  D',  and  from 
the  points  ra',  Z',  &'  to  D'.  Draw  rays  from  the  points  a.  1).  c.  d.  e.f.  g.  h. 
to  the  station  point  S,  and  project  their  intersection  with  the  base  lines  to 
the  perspective  line  a'  D'  as  in  previous  examples :  the  intersection  of  the 
lines  Jc'  D7, 1'  D',  m'  D7  by  the  perpendiculars  thus  projected,  will  establish 
the  points  of  the  curve  of  the  arch  on  the  side  nearest  the  spectator.  To 
determine  the  position  of  the  opposite  side  of  the  arch,  from  a",  the  per- 
spective width  of  the  bridge,  draw  a"  D',  and  from  h'  draw  lines  to  D ; 


PERSPECTIVE   DRAWING.  401 

the  line  h''p'  will  be  the  perspective  width  of  the  pier  ;  draw'  V  D ;  and  from 
F,  k*  D ;  from  /  the  intersection  of  the  curve  of  the  arch  by  the  perpen- 
dicular to  /,  draw  /  D,  the  intersection  with  k"  D'  will  be  one  point  in 
the  curve  of  the  arch  on  the  opposite  side  of  the  bridge :  in  the  same  way, 
from  any  point  in  the  nearer  arc  draw  lines  to  D,  and  the  intersection 
with  lines  in  the  same  planes  on  the  opposite  side  of  the  bridge,  will  fur- 
nish points  for  the  further  arch :  all  below  the  first  only  will  be  visible  to 
the  spectator. 

To  Draw  in  Parallel  Perspective  the  Interior  of  a  Room. — PI.  GUI. 
We  propose  to  construct  this  by  scale  without  laying  down  the  plan. 
Draw  the  horizon  line  D  Y  D',  and  the  base  H  M',  making  D  and  D'  the 
point  of  distance.  Let  the  room  be  20  feet  wide,  14  feet  high,  and  12  feet 
deep ;  on  the  base  M  M',  lay  off  the  rectangle  of  the  section  in  our  figure 
on  a  scale  of  8  feet  to  the  inch,  20  feet  x  14  feet.  From  the  four  corners 
draw  lines  to  the  centre  of  view  V ;  from  S'  lay  off  to  the  right  or  left  on 
M  M'  12  feet,  and  through  this  point  draw  lines  to  D'  or  D  as  the  case  may 
be :  through  the  point  of  intersection  a'  of  this  line  with  S'  V  draw  a  line 
parallel  to  M  M' ;  at  the  intersections  of  this  line  with  M  Y  and  M'  Y  erect 
a  perpendicular,  cutting  the  vanishing  lines  of  the  upper  angle  of  the 
room  at  d  and  e  /  connect  d  e  and  the  perspective  of  the  room  is  complete. 
To  draw  the  aperture  for  a  door  or  window  on  the  side,  measure  off  from 
S'  the  distance  of  the  near  side  from  the  plane  of  the  picture,  and  in  addi- 
tion thereto  the  width  of  the  aperture  ;  from  these  two  points  draw  lines  to 
the  proper  point  of  distance,  and  at  their  intersection  with  S'  Y,  draw  par- 
allels to  M  M',  cutting  the  lower  angles  of  the  room,  and  erect  perpendic- 
ulars, the  height  of  which  will  be  determined  by  a  line  drawn  from/,  the 
height  of  the  window  above  the  floor  measured  on  M  D.  Should  the  win- 
dow be  recessed,  the  farther  jamb  will  be  visible ;  extend  the  farther  par- 
allel to  M  M',  and  cut  it  by  a  line  g  Y.  M  g  being  the  depth  of  the  recess, 
the  rest  of  the  construction  may  be  easily  understood  by  inspection  of  the 
figure.  At  the  extremity  of  the  apartment  a  door  is  represented  half  open, 
hence  as  the  plane  of  the  door  is  at  right  angles  to  the  plane  of  the  picture, 
the  top  and  bottom  lines  will  meet  in  the  point  of  view  ;  if  the  door  were 
open  at  an  angle  of  45°,  these  lines  would  meet  in  the  points  of  distance ; 
if  at  any  other  angle,  the  vanishing  points  would  have  to  be  determined 
by  constructing  a  plan,  drawing  a  line  parallel  to  the  side  of  the  door 
through  the  station  point,  and  projecting  it  upon  the  horizon  line.  The 
chair  in  the  middle  of  the  room  is  placed  diagonally,  and  the  table  parallel 
to  the  plane  of  the  picture ;  their  projection  is  simple. 

To  Draw  in  Perspective  a  Flight  of  Stairs.— PL  CIV.  Lay  off  the 
26 


402  PEESPECTIVE   DKAWING. 

base  line,  horizon,  centre  of  view,  and  point  of  distance  of  the  picture, 
construct  the  solid  abed,  efg  A,  containing  the  stairs,  and  in  the  required 
position  in  the  plane  of  the  picture,  divide  the  rise  a  c  into  equal  parts  ac- 
cording to  the  number  of  stairs,  four  for  instance ;  divide  perspectively  the 
line  a  b  into  the  same  number  of  parts  ;  at  the  points  of  division  of  this 
latter  erect  perpendiculars,  and  through  the  former  draw  lines  to  the  cen- 
tre of  view ;  one  will  form  the  rise  and  the  other  the  tread  of  the  steps. 
From  the  top  of  the  first  step  to  the  top  of  the  upper  continue  a  line  a  d, 
till  it  meets  the  perpendicular  S'  Y  prolonged  in  v  /  this  line  will  be  the 
inclination  or  pitch  of  the  stair ;  if  through  the  top  of  the  step  at  the  other 
extremity  a  similar  line  be  drawn,  it  will  meet  the  central  perpendicular 
at  the  same  point  v,  and  will  define  the  length  of  the  lines  of  nosing  of 
the  steps,  and  the  other  lines  may  be  completed.  As  the  pitch  lines  of  both 
sides  of  the  stairs  meet  the  central  vertical  -in  the  same  point,  in  like  man- 
ner v  will  be  the  vanishing  point  of  all  lines  having  a  similar  inclination 
to  the  plane  of  the  picture.  The  projection  of  the  other  flight  of  stairs 
will  be  easily  understood  from  the  lines  of  construction  perpendicular  to 
the  base  line  or  parallel  thereto,  lying  in  planes. 

To  Find  the  Reflection  of  Objects  in  the  Water. — PI.  CIY.  Let  B  be 
a  cube  suspended  above  the  water ;  we  find  the  reflection  of  the  point  a, 
but  letting  fall  a  perpendicular  from  it,  and  setting  off  the  distance  af  w 
below  the  plane  of  the  water  equal  to  the  line  a  w  above  this  line  ;  the  line 
wf  will  also  be  equal  to  the  line  wf;  find  in  the  same  way  the  points  V 
and  e',  through  these  points  construct  perspectively  a  cube  in  this  lower 
plane,  and  we  have  the  reflection  of  the  cube  above. 

To  find  the  reflection  of  the  square  pillar  D  removed  from  the  shore  : 
suppose  the  plane  of  the  water  extended  beneath  the  pillar,  and  proceed 
as  in  the  previous  example. 

It  will  be  observed  that  those  lines  of  an  object  which  meet  in  the  cen- 
tre of  view  Y,  in  the  original ;  their  corresponding  reflected  lines  will  con- 
verge to  the  same  point.  If  the  originals  converge  to  the  points  of  distance, 
the  reflected  ones  will  do  the  same.  To  find  the  reflection  of  any  inclined 
line,  find  the  reflection  of  the  rectangle  of  which  it  is  the  diagonal,  if  the 
plane  of  the  rectangle  is  perpendicular  to  the  plane  of  the  picture  ;  if  the 
line  is  inclined  in  both  directions  enclose  it  in  a  parallelopided  and  project 
the  reflection  of  the  solid. 

To  find  the  Perspective  Projection  of  Shadows. — Plate  CY.  Let  the 
construction  points  and  lines  of  the  picture  be  plotted.  Let  A  be  the  per- 
spective projection  of  a  cube  placed  against  another  block,  of  which  the 
face  is  parallel  to  the  plane  of  the  picture :  to  find  the  shadow  upon  the 


PERSPECTIVE   DRAWING.  403 

block  and  upon  the  ground  plane,  supposing  the  light  to  come  into  the 
picture  from  the  upper  left-hand  corner  and  at  an  angle  of  45°.  Since  the 
angle  of  light  is  the  diagonal  of  a  cube,  construct  another  cube  similar  to 
A,  and  adjacent  to  the  face  d  c  g ;  draw  the  diagonal  Ik,  it  will  be  the 
direction  of  the  rays  of  light,  and  Jc  will  be  the  shadow  of  I;  connect/7; 
and  c7c,fk  must  be  the  shadow  of  the  line  If,  and  ck  of  b  c;  the  one 
upon  the  horizontal  plane  and  the  other  in  a  vertical  one :  the  former  will 
have  its  direction,  being  a  diagonal,  toward  the  point  of  distance  D',  the 
other  being  a  diagonal  in  a  plane,  parallel  to  that  of  the  picture,  will  be 
always  projected  upon  this  plane  in  a  parallel  direction. 

Let  B  be  a  cube  similar  to  A ;  to  find  its  projection  upon  a  horizontal 
plane,  the  shadow  of  the  point  V  may  be  determined  as  in  the  preceding 
example,  but  the  shadow  of  the  point  c',  instead  of  falling  upon  a  plane 
parallel  to  the  picture,  falls  upon  a  horizontal  one ;  its  position  must  be  de- 
termined as  we  did  before  by  b.  Construct  the  cube  and  draw  the  diag- 
onal c'l;  in  the  same  way  determine  the  point  in1  the  shadow  of  df  ;  con- 
nect c  k'  I  m  n,  and  we  have  the  shadow  of  the  cube  in  perspective  on  a 
horizontal  plane. 

On  examination  of  these  projected  shadows,  it  will  be  found  that  as  the 
rays  of  light  fall  in  a  parallel  direction  to  the  diagonal  of  the  cube,  the 
vanishing  point  of  these  rays  will  be  in  one  point  V  on  the  line  D'  M' 
prolonged,  at  a  distance  below  D'  equal  V  D' ;  and  since  the  shadows  of 
vertical  lines  upon  a  horizontal  plane  are  always  directed  towards  the  point 
of  sight,  the  extent  of  the  shadow  of  a  vertical  line  may  be  determined 
by  the  intersection  of  the  shadow  of  the  ground  point  of  the  line  by  the 
line  of  light,  from  the  other  extremity.  Thus,  the  point  k,  cube  A,  is  the 
intersection  of/  D'  by  5  V ;  the  points  k',  I,  m  are  the  intersections  of  c  D', 
o  D',  n'  D'  by  V  Y7,  G'  Y'by  df  Y.'  Similarly  on  planes  parallel  to  that  of  the 
picture,  k,  cube  A  is  intersection  of  the  diagonal  c  k,  by  the  ray  of  light  5  V. 

Applying  this  rule  to  the  frame  0,  fromr,  s,p,  draw  lines  to  D'  from  /, 
s '  iP ' •>  draw  rays  to  Y';  their  intersections  define  the  outline  of  the  shadow  of 
the  post.  To  draw  the  shadow  of  the  projection,  the  shadow  upon  the  post 
from  t  will  follow  the  direction  of  the  diagonal  G  k.  Project  u  and  v  upon 
the  ground  plane  at  u'  andv'y  from  t'  u'  v'  andj?  draw  lines  to  D';  from  t, 
u,  v,  w  and  x  draw  rays  to  Y',  and  the  intersection  of  these  lines  with  their 
corresponding  lines  from  their  bases  will  give  the  outline  required ;  as  v  and  w 
are  on  the  same  perpendicular,  their  rays  will  intersect  the  same  line  v'  Y'. 

With  reference  to  the  intensity  of."  shade  and  shadow  "  and  the  neces- 
sary manipulation  to  produce  the  required  eifect,  the  reader  is  referred  to 
the  article  on  this  subject. 


404  PERSPECTIVE  DRAWING. 

In  treating  of  Perspective  it  has  been  considered  not  in  an  artistic 
point,  as  enabling  a  person  to  draw  from  nature,  but  rather  as  a  useful  art  to 
assist  the  architect  or  engineer  to  complete  his  designs,  by  exhibiting  them 
in  a  view  such  as  they  would  have  to  the  eye  of  a  spectator  when  con- 
structed. In  our  examples,  owing  to  size  of  the  page,  we  have  been  limit- 
ed in  the  scale  of  the  figures,  and  in  the  distance  of  the  point  of  view,  or 
distance  of  the  eye  from  the  plane  of  the  picture,  and  as  it  was  unimportant 
to  the  mathematical  demonstration,  few  of  the  figures  extend  above  the 
line  of  the  horizon.  In  these  particular  points  it  is  unnecessary  that  the 
examples  should  be  copied.  The  most  agreeable  perspective  representa- 
tions are  generally  considered  to  be  produced  by  fixing  the  angle  of  vision 
M  S  M',  at  from  45  to  50°,  and  the  distance  of  the  horizon  above  the 
ground  line  at  about  one-third  the  height  of  the  picture. 

Linear  perspective  is  more  adapted  to  the  representation  of  edifices, 
bridges,  interiors,  &c.,  than  to  that  of  machinery ;  it  belongs,  therefore, 
rather  to  the  architect  than  to  the  engineer  or  the  mechanic ;  for  the  pur- 
poses of  the  latter  we  would  recommend  Isometrical  Perspective,  uniting 
accuracy  of  measures  with  graphic  perspective  representation. 


ISOMETRICAL   DRAWING.  4Q5 


ISOMETKICAL  DKAWING. 

PROFESSOR  PARISH,  of  Cambridge,  has  given  the  term  Isometrical  Per- 
spective to  a  particular  projection  which  represents  a  cube,  as  in  fig.  1. 
The  words  imply  that  the  measure  of  the  representations  of  the  lines 
forming  the  sides  of  each  face  are  equal. 

The  principle  of  isometric  representation  con- 
sists in  selecting  for  the  plane  of  the  projection, 
one  equally  inclined  to  three  principal  axes,  at 
right  angles  to  each  other,  so  that  all  straight  lines 
coincident  with  or  parallel  to  these  axes,  are  drawn 
in  projection  to  the  same  scale.  The  axes  are 
called  isometric  axes,  and  all  lines  parallel  to  them 
are  called  isometric  lines.  The  planes  containing 
the  isometric  axes  are  isometric  planes ;  the  point  Fi&- 

in  the  object  projected,  assumed  as  the  origin  of  the  axes,  is  called  the  reg- 
ulating point. 

To  draw  the  isometrical  projection  of  a  cube,  (fig.  2,)  draw  the  hori- 
zontal line  A  B  indefinitely  ;  at  the  point  D  erect  the  perpendicular  D  C, 
equal  to  one  side  of  the  cube  required;  through  D  draw  the  line  D5  and. 
T)f  to  the  right  and  left,  making  f  D  B  and  5  D  A  each  equal  an  angle  of 
30°.  Consequently  the  angles  F  ~Df  and  F  D  5  are  each  equal  to  60°. 
Make  D  1)  and  ~Df  each  equal  to  the  side  of  the  cube,  and  at  b  and/"  erect 
perpendiculars,  making  5  a  and  f  e  each  equal  to  the  side  of  the  cube ; 
connect  F  a  and  F  e  and  draw  e  g  parallel  to  a  F,  and  a  g  parallel  to  F  <?, 
and  we  obtain  the  projection  of  the  cube. 

If  from  the  point  F,  with  a  radius  F  D,  a  circle  be  described,  and  com- 
mencing at  the  point  D  radii  be  laid  off  around  the  circumference,  forming 
a  regular  inscribed  hexagon,  and  the  points  D  a  e  be  connected  with  the 


406 


ISOMETRICAL   DRAWING. 


centre  of  the  circle  F,  we  have  an  isometrical  representation  of  a  cube. 
The  point  D  is  called  the  regulating  point. 

If  a  cube  be  projected  according  to  the  principles  of  isometrical  per- 
spective, in  a  similar  manner  as  we  have  constructed  one  according  to  the 
rules  of  linear  perspective,  the  length  of  the  isometrical  lines  would  be  to 
the  original  lines  as  .8164  to  1,  but  since  the  value  of  isometrical  perspec- 
tive as  a  practical  art  lies  in  the  applicability  of  common  and  known 
scales  t6  the  isometric  lines,  in  our  constructions  we  have  not  thought  it 
necessary  to  exemplify  the  principles  of  the  projection,  but  have  drawn  our 
figures  without  any  reference  to  what  would  be  the  comparative  size  of  the 
original  and  of  the  projection,  transferring  measures  directly  from  plans  and 
elevations  in  orthographic  projections,  to  those  in  isometry.  It  will  be  ob- 
served that  the  isometric  scale  adopted  applies  only  to  isometric  lines,  as 
F  D,  F  a  and  F  e  or  lines  parallel  thereto ;  the  diagonals  which  are  abso- 
lutely equal  to  each  other,  and  longer  than  the  sides  of  the  cube,  are  the 
one  less, the  other  greater  ;  the  minor  axis  being  unity,  the  isometrical  lines 
and  the  major  axis  are  to  each  other  as,  1.  -j/2.  y3. 

Understanding  the  isometrical  projection  of  a  cube,  any  surface  or 
solid  may  be  similarly  constructed,  since  it  is  easy  to  suppose  a  cube  suffi- 
ciently large  to  contain  within  it  the  whole  of  the  model  intended  to  be 
represented,  and  as  hereafter  will  be  farther  illustrated,  the  position  of  any 
point  on  or  within  the  cube,  the  direction  of  any  line  or  the  inclination  of 
any  plane  to  which  it  may  be  cut,  can  be  easily  ascertained  and  repre- 
sented. 

In  figs.  1  and  2  one  face  of  the 
cube  appears  horizontal,  and  the 
other  two  faces  appear  vertical.  If 
now  the  figures  be  inverted,  that 
which  before  appeared  to  be  the  top 
of  the  object,  will  now  appear  to 
be  its  under  side. 

The  angle  of  the  cube  formed 
by  the  three  radii  meeting  in  the 
centre    of    the    hexagon,   may  be 
made  to  appear  either  an  internal  or 
external  angle ;  in  the  one  case  the 
'""    faces  representing  the  interior,  and 
in  the  other  the  exterior  of  a  cube. 
Figs.  3,  4,  5,  illustrate  the  application  of  isometrical  drawing  to  simple 


ISOMETRICAL   DRAWING.  407 

combinations  of  the  cube  and  parallelopipedon.    The  mode  of  construction 


Figs.  3,  4,  5. 


of  these  figures  will  be  easily  understood  by  inspection,  as  they  contain  no 
lines  except  isometrical  ones. 

To  draw  Angles  to  the  Boundary  Lines  of  an  Isometrical  Cube. 


40     SO    20    10 


Fig.  6,  7. 


Draw  a  square  C  (fig.  6,)  whose  sides  are  equal  to  those  of  the  isome- 
trical cube  A,  and  from  any  of  its  angles  describe  a  quadrant,  which  di- 
vide into  90°,  and  draw  radii  through  the  divisions  meeting  the  sides  of 
the  square.  These  will  then  form  a  scale  to  be  applied  to  the  faces  of  the 
cube ;  thus  on  D  E,  or  any  other,  by  making  the  same  divisions  along  their 
respective  edges. 

As  the  figure  has  twelve  isometrical  sides,  and  the  scale  of  tangents 


408 


ISOMETRICAL   DE AWING. 


may  be  applied  two  ways  to  each,  it  can  be  applied  therefore  twenty-four 
ways  in  all.  "We  thus  have  a  simple  means  of  drawing,  on  the  isometrical 
faces  of  the  cube,  lines,  forming  any  angles  with  their  boundaries. 

Figs.  1,  2,  3,  4,  5,  6,  PL  CVI.  show  the  section  of  a  cube  by  single 
planes,  at  various  inclinations  to  the  faces  of  the  cubes.  Figs.  7,  8  are  the 
same  cube,  but  turned  round,  with  pieces  cut  out  of  it.  Fig.  9  is  a  cube 
cut  by  two  planes  forming  the  projection  of  a  roof.  Fig.  10  is  a  cube  with 
all  of  the  angles  cut  off  by  planes,  so  as  to  leave  each  face  an  octagon. 
Fig.  11,  represents  the  angles  cut  off  by  planes  perpendicular  to  the  base 
of  the  cube,  forming  thereby  a  regular  octagonal  cylinder.  By  drawing 
lines  from  each  of  the  angles  of  an  octagonal  base  to  the  centre  point  of 
the  upper  face  of  the  cube,  we  have  the  isometrical  representation  of  an 
octagonal  prism. 

As  the  lines  of  construction  have  all  been  retained  in  these  figures, 
they  will  be  easily  understood  and  copied,  and  are  sufficient  illustrations 
of  the  method  of  representing  any  solid  by  enclosing  it  in  a  cube. 

We  have  now  to  consider  the  application  of  this  species  of  projection 
to  curved  lines. 


x 


Figs.  8,  9. 


Let  A  B  (fig.  8,)  be  the  side  of  a  cube  with  a  circle  inscribed ;  and 
suppose  all  the  faces  of  the  cube  to  have  similarly  inscribed  circles.  Draw 
the  diagonals  A B,  CD,  and  at  their  intersection  with  the  circumference 
lines  parallel  to  A  C,  B  D.  Now  draw  the  isometrical  projection  of  the 
cube,  (fig.  9,)  and  lay  out  on  the  several  faces  the  diagonals  and  the  par- 
allels; the  projection  of  the  circle  will  be  an  ellipse,  of  which  the  diago- 
nals being  the  axes,  their  extremities  are  defined  by  their  intersections/ 6, 


ISOMETRICAL   DRAWING. 


409 


e  5,  a  2,  1 1,  ^3,  c  4,  by  the  parallels;  having  thus  the  major  and  minor 
axis,  construct  the  ellipse  by  the  trammel,  or  since  the  curve  is  tangent  at 
the  centre  of  the  sides,  we  have  eight  points  in  the  curve ;  it  may  be  put 
in  by  sweeps  or  by  the  hand. 


Fig.  10. 

To  Divide  the  Circumference  of  a  Circle. — First  method.  'On  the 
centre  of  the  line  A  B  erect  a  perpendicular  C  D,  making  it  equal  to  C  A 
or  C  B ;  then  from  D,  with  any  radius,  describe  an  arc  and  divide  it  in 
the  ratio  required,  and  draw  the  divisions  radii  from  D  meeting  A  B ; 
then  from  the  isometric  centre  of  the  circle  draw  radii  from  the  divisions 
on  A  B,  cutting  the  circumference  in  the  points  required. 

Second  method.  On  the  major  axis  of  the  ellipse  describe  a  semi-cir- 
cle, and  divide  it  in  the  manner  required.  Through  the  points  of  division 
draw  lines  perpendicular  to  A  E,  which  will  divide  the  circumference  of 
the  ellipse  in  the  same  ratio.  On  the  right  hand  of  the  figure  both 
methods  are  shown  in  combination,  and  the  intersection  of  the  lines  give 
the  points  in  the  ellipse. 

Plate  CYII.  The  upper  figure  is  an  isometrical  projection  of  a  bevel 
wheel,  with  a  half-plan  beneath,  and  projected  lines  explanatory  of  the 
method  to  be  adopted  in  drawing  the  teeth,  and  of  which  only  half  are 
shown  as  cut.  It  will  be  seen  by  reference  to  the  second  method  given 
above  for  the  division  of  the  circumference  of  a  circle,  that  the  semicircle 
is  described  directly  on  the  major  axis  of  the  ellipse ;  in  practice  it  will  be 
found  more  convenient,  when  a  full  drawing  is  to  be  made,  to  draw  the 
semicircle  on  a  line  parallel  to  the  major  axis,  and  entirely  without  the 


410 


ISOMETKICAL   DRAWING. 


lines  of  the  main  drawing.  And  also,  as  in  the  example  of  the  bevel  gear, 
complete  on  the  semicircle,  or  half-plan,  the  drawings  of  all  lines,  the 
intersection  of  which  with  circles  will  be  necessary  to  be  projected  on  the 
isometrical  drawing. 

The  lower  figure  is  an  isometrical  projection  of  a  complete  pillow 
block,  with  its  hold-down  bolts.  By  reference  to  Plate  XXII.  and 
figs.  238  and  239,  p.  146,  it  will  be  seen  how  much  more  graphically 
these  forms  of  gearing  are  given  by  isometry  than  by  the  usual  projec- 
tion. As  an  exercise  for  the  learner,  it  will  be  very  good  practice  to 
project  isometrically  the  spur-gear,  Plate  XVIII.,  and  the  standard  and 
hanger,  Plates  XV.  and  XVI.,  of  which  sufficient  details  are  given  in 
the  plates. 

Plate  CVIIL  is  an  isometrical  projection  of  a  culvert,  such  as  were 
built  beneath  the  Croton  Aqueduct,  and  is  a  good  example  of  construc- 
tion, and  better  illustrated  by  the  drawing  than  it  would  be  by  plan  and 
elevations. 


Fig.  11. 


Figs.  11, 12,  and  13  are  further  examples  to  show  the  applicability,  of 
isometry  to  other  forms  of  construction. 


•  ISOMETRICAL   DKAWIXG.  411 

Fig.  11  is  a  sectional  view  of  a  very  complicated  boiler,  the  construc- 
tion of  which  could  hardly  be  explained  to  a  mechanic  by  any  number  of 
sectional  drawings,  and  only  by  isometry  or  perspective  to  one  not  con- 
versant with  such  matters. 


Fig.  12. 

Fig.  12  is  an  elevation  of  a  portion  of  a  truss  of  an  iron  bridge,  given 
not  as  an  example  of  construction,  but  merely  of  the  projection. 

Fig.  13  is  an  elevation  of  the  roof-truss,  fig.  2,  Plate  XL VI. ;  no  side 
view  is  shown  on  the  plate,  but  the  dimensions  of  timber  and  spaces  are 
drawn  as  usual  in  practice. 

Plate  CIX.  is  an  elevation  and  section  in  isometry  of  the  district  school- 
house  given  in  Plate  LXXIV.  The  scale  has  been  necessarily  reduced  to 
bring  the  drawing  within  the  limits  of  the  page,  but  is  given  on  the  plate 
as  it  should  always  be  either  drawn  or  written,  on  all  drawings  to  a  scale, 
not  intended  for  mere  pictures  or  illustrations.  The  section  is  drawn  at 
the  height  of  8  feet  above  the  base  course,  and  higher  than  is  usual  in  such 
sections,  but  it  was  necessary  on  account  of  the  extra  height  of  the  window- 
sill  above  the  floor,  desirable  in  all  school-rooms.  As  a  plan  it  is  more 
graphic  than  that  given  in  Plate  LXXIV.,  and  when  there  are  staircases 
one  above  the  other  in  the  drawing,  they  are  more  intelligibly  expressed, 
but  there  is  nothing  in  the  present  drawing  that  cannot  be  nearly  as  well 
shown  by  the  plan ;  and  to  a  mechanic,  for  the  purposes  of  construction,  the 
plan  is  the  simpler. 


412 


ISOMETRICAL   DRAWING. 


By  comparing  the  elevation,  Plate  CIX.,  with  the  perspective,  GIL, 
the  former  appears  distorted,  and  out  of  drawing,  but  it  is  much  more 
readily  drawn,  and  has  this  great  convenience,  that  it  is  drawn  and  can  be 
measured  by  a  scale,  but  only  on  the  isometric  lines — all  others  are  dis- 


torted,  too  long  or  too  short,  as  may  be  seen  in  the  major  and  minor  axes 
of  the  bevel  gear,  Plate  CYII.,  or  the  rake  lines  of  the  roof,  Plate  CIX. 

We  have  multiplied  examples  of  isometrical  drawing,  to  show  its  ap- 
plicability to  varied  forms  of  construction,  mechanical  and  architectural. 
The  principles  of  this  projection  are  easy  and  intelligible,  and  their  use 
should  be  extended.  Isometrical  projection  is  especially  valuable  to  the 
mechanical  draughtsman,  explaining  many  constructions  that  could  hardly 
be  done  by  any  amount  of  plans,  elevations,  and  sections,  and  still  uniting 
with  pictorial  representation  the  applicability  of  a  scale.  For  drawings 
for  the  Patent  Office  it  is  especially  desirable,  in  a  simple  and  practical 
form  combining  the  requisites  of  many  projections  ;  but  as  a  drawing  of 


WINDLASS. 


413 


414 


ISOMETEICAL   DRAWING. 


what  could  be  absolutely  seen  by  the  eye  it  is  not  truthful,  and  therefore 
when  pictorial  illustration  only  is  requisite,  the  drawing  should  be  in  linear 
perspective. 

In  confirmation  of  the  above,  we  give  below,  and  on  page  413,  two 
drawings  in  perspective,  in  which  the  point  of  sight  is  above  the  plane 
of  the  picture,  and  approaching  in  general  appearance  to  drawings  in 
isometry ;  and  yet,  having  all  the  truthfulness  of  sight,  are  much  better 
suited  to  the  purposes  for  which  they  were  intended — illustrations  of  a 
specimen  book  of  Ship-work,  as  manufactured  by  James  L.  Jackson  & 
Brother,  of  this  city. 


Centre-Board  Winch. 


ENGINEEKING  DKAWESTG.  415 


ENGINEERING  DRAWING. 

TEEDGOLD  defines  "  civil  engineering  as  the  art  of  directing  the  great 
sources  of  power  in  nature  for  the  use  and  convenience  of  man,  being  the 
practical  application  of  the  most  important  principles  of  natural  philoso- 
phy. *  The  most  important  object  of  civil  engineering  is,  to  im- 
prove the  means  of  production  and  of  traffic  in  States,  both  for  external 
and  internal  trade.  It  is  applied  to  the  construction  and  management  of 
roads,  bridges,  railroads,  aqueducts,  canals,  river  navigation,  docks,  and 
storehouses,  for  the  convenience  of  internal  intercourse  and  exchange;  and 
in  the  construction  of  ports,  harbors,  moles  and  breakwaters,  and  light- 
houses; and  in  the  navigation  by  artificial  power  for  the  purposes  of 
commerce. 

"  Besides  these  great  objects  of  individual  and  national  interest,  it  is 
applied  to  the  protection  of  property  where  natural  powers  are  the  sources 
of  injury,  as  by  embankments  for  the  defence  of  tracts  of  country  from 
the  encroachments  of  the  sea  or  the  overflowing  of  rivers ;  it  also  directs 
the  means  of  applying  streams  and  rivers  to  use  either  as  powers  to  work 
machines  or  as  supplies  for  the  use  of  cities  and  towns,  or  for  irrigation,  as 
well  as  the  means  of  removing  noxious  accumulations,  as  by  the  drainage 
of  towns  and  districts  to  prevent  the  formation  of  malaria,  and  secure  the 
public  health.  This  is  only  a  brief  sketch  of  the  objects  of  civil  engineer- 
ing ;  the  real  extent  to  which  it  may  be  applied  is  limited  only  by  the 
progress  of  science ;  its  scope  and  utility  will  be  increased  with  every  dis- 
covery in  philosophy,  and  its  resources  with  every  invention  in  mechanical 
or  chemical  art." 

The  object  of  the  engineer  should  be  to  arrange  his  material  with  re- 
gard to  economical  effects.  The  artistic  is  considered  to  be  the  depart- 
ment of  the  architect,  who  was  originally,  as  the  term  implies,  the  chief 
builder.  It  is  not  indispensable  that  the  engineer  should  know  any  thing 
of  architecture  in  its  signification  of  a  fine  art — though  it  would  be  better, 
in  many  instances,  if  he  did ;  when  properly  carried  out,  all  his  works 


416  ENGINEERING  DRAWING. 

should  show  that  beauty  which  is  inseparable  from  truthfulness  of  design 
and  fitness  for  purpose.  The  engineer,  according  to  Ferguson,  should  be 
"  the  architect  who  occupies  himself  more  especially  with  construction  and 
the  more  utilitarian  class  of  works,  and  the  architect  the  artist  who  at- 
tends to  the  ornamental  distribution  of  buildings,  and  their  decoration 
when  erected." 

There  is  little  in  the  present  volume,  except  the  artistic  portion  of 
Architectural  Drawing,  that  might  not  well  come  under  the  general  head 
of  Engineering  Drawing ;  and  it  is  not,  therefore,  proposed  to  repeat  what 
has  already  been  treated  of,  but  to  illustrate  in  a  similar  way  other  depart- 
ments, especially  of  hydraulic  and  railway  engineering,  with  numerous 
examples,  drawn  mostly  from  practice  in  this  country,  reference  being  had 
to  the  rules  and  principles  of  construction,  before  given,  with  such  addi- 
tions as  may  seem  necessary. 

There  is  no  branch  of  engineering  more  important  than  the  securing 
of  a  good  foundation,  and  none  so  difficult  for  the  application  of  rules. 
The  general  requirements  are  stated  in  the  beginning  of  Architectural 
Drawing,  and  the  expedients  commonly  adopted,  but  the  importance  of 
the  subject  demands  further  illustration. 

Piles  are  used  either  as  posts  or  columns  driven  through  soft  earth 
to  a  hard  bottom,  or  depending  on  their  exterior  frictional  surface  to  give 
the  necessary  support,  either  in  earth  naturally  compact  or  made  so  by 
the  driving  of  the  piles.  In  the  first  case,  care  must  be  taken  that  the 
piles  be  driven  sufficiently  deep  into  the  lower  strata  to  secure  their  ends 
from  slipping  laterally,  and  soundings  should  be  made  carefully  to  ascer- 
tain the  dip  and  character  of  this  strata.  In  many  places,  from  the  hard- 
ness and  the  inclined  position  of  the  lower  strata,  this  kind  of  foundation  is 
inapplicable  and  unsafe. 

In  the  case  in  which  the  support  from  the  piles  depends  on  the  exterior 
frictional  resistance,  the  rule  most  generally  adopted  by  engineers  is  that 
of  Major  Saunders,  published  in  the  Journal  of  the  Franklin  Institute  for 
1851: 

Multiply  the  weight  of  the  ram  by  the  distance  which  the  ram  falls,  in 
inches,  at  last  blow,  divided  by  8  times  the  depth  driven  or  set  at  that 
blow.  Thus,  suppose  the  ram  to  be  1600  Ibs.  weight,  the  fall  20  ft.,  or  240*, 

and  the  set  \  inch,  then  the  safe  load  would  be  -. — 5- — =96,000  Ibs. 

2"X  O 

According  to  Claudel,  the  weight  on  a  pile  should  not  exceed  about 
800  Ibs.  to  the  square  inch.  The  above  pile  should  therefore  be  A|^^JL= 
120  square  inches,  or  say  12  inches  diameter. 


ENGINEERING   DRAWING. 


41T 


The  usual  weight  of  the  ram,  or  hammer,  employed  on  our  public  works, 
varies  from  1,400  to  2,400  Ibs.,  and  the  height  of  leaders,  or  fall,  from 
20  to  35  ft.  Many  have  been  made  much  higher ;  but  Mr.W.  J.  McAlpine 
states,  as  the  result  of  his  experiments,  that  "  there  is  no  increased  force  of 
blow  obtained  by  a  fall  of  more  than  40  feet,  as  the  friction  on  the  ways  is 
increased  so  rapidly  that  no  increased  velocity  is  obtained  by  falling  from  a 
greater  height."  The  leaders  required  to  drive  piles  on  N.  Y.  State  canals 
were  to  be  35  feet  high,  and  weight  of  hammer  not  less  than  1,600  Ibs. 
But  there  is  an  advantage  gained  by  increasing  the  weight  of  the  hammer 
and  reducing  the  fall ;  there  is  less  damage  done  to  the  head  of  the  pile 
and  to  the  iron  rings  or  hoops  for  the  head,  and  the  more  frequent  the 
blows  the  more  readily  the  pile  is  driven.  Piles  were  driven  by  the  Nas- 
myth  machine  in  7  minutes,  while  an  hour  was  required  for  a  similar  pile 
by  another  machine. 

In  specifications  piles  are  usually  required  to  be  smooth  and  straight, 
and  at  least  10  inches  diam.  at  the  lower  end,  and  that  there  should  be  only 
so  much  set,  say  an  average  of  1"  per  blow  for  the  last  4  blows.  This  last 
will  be  left  somewhat  to  the  discretion  of  the  engineer  after  watching  the 
driving  a  few  of  the  piles.  Piles  driven  beneath  the  surface  of  the  ground, 
and  always  kept  wet,  may  be  of  any  timber — that  is  straight  enough,  and 
will  stand  the  blows  of  the  hammer — but  exposed  .piles  should  be  either 
oak  or  chestnut. 

As  usually  driven,  and  of  average  size,  when  the  whole  weight  is  to  be 
supported  by  the  pile,  10  tons  may  be  reckoned  as  a  safe  load ;  but  where 
additional  support  is  received  from  the  compacted  earth,  or  from  concrete, 


Fig.  1.  Fig.  2. 

it  would  be  impossible  to  assign  a  limit  based  on  the  pile  alone ;  to  those 
under  the  bridge  of  Neuilly,  there  is  a  load  of  57  tons  for  each  pile. 

Sheet-piling  (figs.  1,  2)  is  usually  of  plank  2  to  3  inches  thick,  set  or 

27 


418 


ENGINEERING   DRAWING. 


driven.  For  driving,  the  bottom  of  the  plank  should  be  sharpened  to  a 
chisel-edge,  a  little  out  of  centre  toward  the  timber  side,  and  cornered 
slightly  at  the  outer  edge,  that  it  may  hug  the  timber  and  the  plank  before 
driven.  I  have  driven  successfully,  by  a  steam  pile-driver,  timber  sheet- 
piling  6  inches  thick,  and  from  25  to  30  feet  long.  The  necessities  of  the 
case  required  prompt  action — timber  was  taken  of  such  variety  and  such 
width  as  could  be  procured — each  pile,  of  course,  being  uniform  in  width, 
but  varying  from  6"  to  18"  for  different  piles.  Southern  pine  was  found  to 
drive  the  best.  At  first  each  pile  was  grooved  V  x  Z"  (fig.  3),  and  2"  x  1" 
slip-tongue  inserted,  and  strongly  spiked,  but  as  time  pressed,  the  groove 
was  formed  by  2"  strips  planted  on. 

Hollow  cast-iron  piles  have  been  driven  by  exhausting  the  air 'from 
the  inside ;  then  the  weight  of  the  pile,  and  sometimes  an  added  load, 
cause  the  pile  to  settle  into  the  earth :  this  is  called  the  vacuum  process. 
The  process  by  plenum  is  by  expelling  the  water  out  of  the  pile  by  forcing 

in  air  in  excess  of  the  pressure  of  the 
surrounding  water,  and  the  workmen 
descending  within  the  pile  and  excavat- 
ing the  material.  This  last  process  was 
adopted  by  Mr.  McAlpine  for  the  piers 
at  the  Harlem  River  bridge.  A  section 
of  a  pile  is  given  (fig.  4).  It  consists 
of  a  series  of  hollow  cylinders  6  ft.  in 
diameter,  9  ft.  longxli-"  thick,  bolted 
;J  together  by  flanges  in  the  inside.  On 
^  the  top  of  the  pile  is  fastened  a  wrought- 
H  iron  cylinder  (a),  with  a  cast-iron  head, 
called  the  air-lock.  In  the  top  and 
bottom  head  there  are  man-holes  (b  b) 
which  can  be  closed  at  pleasure  by  plates 
opening  by  hinges  on  the  lower  sides, 
and  lined  with  rubber  at  the  joints;  in 
the  same  head  there  are  also  two  cocks, 
2"  diam.  Leading  from  the  outside  of 
the  air-lock,  near  its  bottom,  are  two 
curved  tubes  (c  c),  4"  diameter,  passing 
through  the  lower  head,  or  diaphragm, 
of  air-lock,  and  are  closed.by  cocks.  To 
one  of  these  tubes  small  air-pumps  are 
Fig  4  connected  by  flexible  hose.  After  shut- 


.       ENGINEERING   DRAWING.  419 

ting  the  man-hole  in  the  diaphragm,  and  starting  on  the  pumps,  the  pressure 
in  the  pile  commences  to  increase,  and  the  water  is  forced  out  beneath  the 
bottom,  till  the  pressure  is  equal  to  that  due  to  the  head  of  water  outside, 
and  the  water  is  all  out.  The  workmen  now  enter  the  air-lock,  close  the 
man-hole  in  upper  head,  open  the  2"  cock  in  the  diaphragm,  and  equalize 
the  pressure  in  the  air-lock  and  pile.  They  now  open  the  man-hole  in  the 
diaphragm,  and  descend  to  the  bottom  of  the  pile,  and  commence  the  ex- 
cavation of  the  material,  which  is  raised  in  canvas  bags  by  windlass  into 
the  air-lock,  and  removed  at  any  time  outside  by  closing  the  diaphragm 
man-hole,  equalizing  the  air-lock  pressure  with  that  outside,  and  then 
opening  the  upper  head  man-hole.  By  reversing  the  operation,  as  in  the 
descent  of  the  workmen,  material  can  be  taken  into  the  pile. 

When  the  pile  has  been  entirely  cleaned  to  the  bottom,  care  is  taken  to 
see  that  no  obstructions,  such  as  boulders  or  logs,  remain  under  the  edge 
of  the  pile.  The  workmen  then  pass  out  to  the  outer  air  as  did  before  the 
material.  Men  are  then  stationed  at  the  guy -ropes,  the  4"  cock  connec- 
tion with  the  outer  air  is  suddenly  opened,  the  air  rushes  quickly  out  of 
the  pile  and  falls  with  an  effect  like  that  of  a  blow ;  the  water  at  the  same 
time  rushing  in  beneath  the  bottom,  bringing  in  the  earth,  and  leaving  an 
excavation  for  the  settlement  of  the  pile.  The  amount  of  settling  at  one 
time  has  sometimes  amounted  to  12  ft. 

By  repetition  of  this  operation,  the  pile  may  be  sunk  as  far  as  may  be 
required.  It  is  then  to  be  filled  with  concrete  or  masonry.  But  Mr.  McAl- 
pine,  by  driving  under  the  loVer  edge  of  the  pile  (wooden  sheet  piles  5  ft. 
long  x  3"  wide  x  1£  thick)  at  an  angle  of  about  30°,  and  excavating  be- 
neath and  filling  in  with  concrete,  has  managed  to  extend  the  base  very 
considerably.  As  soon  as  the  concrete  or  masonry  has  so  far  progressed 
that  an  effectual  stopper  is  made  at  the  bottom  of  the  pile,  the  air-lock  is 
removed,  and  the  work  is  carried  on  on  the  open  pile. 

When  the  hard  bottom  is  not  very  deep  below  the  level  of  the  water,  it 
is  often  usual  to  curb  out  the  site  of  the  foundation  by  sheet-piling,  or  by 
a  double  curb  of  sheet-piling  set  one  within  the  other,  the  space  between 
the  two  being  filled  in  with  clay  or  some  compact  earth,  or  a  coffer-dam. 
The  water  is  then  kept  pumped  down,  and  the  material  excavated  to  the 
required  depth,  and  the  structure  commenced  and  carried  up.  It  will  be 
necessary  to  brace  the  coffer-dam  or  sheet-piling  curb  in  the  inside,  to  re- 
sist the  pressure  of  water  and  earth  on  the  outside  as  the  work  progresses. 

The  general  plan  adopted  by  Mr.  G.  A.  Parker,  in  the  erection  of  the 
piers  of  the  Susquehanna  bridge,  was  first  to  dredge  away  as  much 
as  possible  of  the  material  in  the  bed  of  the  river  at  the  pier  site.  A  f- "- 


4:20 


ENGINEERING   DRAWING.       • 


thick  boiler-iron  curb  was  then  sunk  and  secured  in  its  place.  The  curb 
was  about  30  ft.  wide  and  50  to  60  ft.  long,  and  of  sufficient  height  to 
reach  above  the  bed  of  the  river.  The  material  was  then  pumped  by  sand- 
pumps  out  of  the  curb,  which  gradually  undermined,  and  settled  down  to 
the  required  depth,  or  on  to  the  bed-rock.  When  stumps,  logs,  or  boulders 
were  met  with,  they  were  removed  by  divers  working  in  a  bell.  After  the 
rock  had  been  thoroughly  cleaned  off,  it  was  brought  to  a  uniform  level  by 
a  solid  bed  of  concrete  extending  over  a  greater  space  than  the  size  of  the 
bottom  of  the  pier,  using  the  diving-bell  for  this  purpose. 

Three  guide-piles  on  each  side,  and  one  at  each  end,  were  fixed  firmly 
in  position.  A  strong  platform  of  solid  timber,  the  size  of  the  bottom  of 
the  pier,  was  then  placed  in  position  over  the  curb,  and  at  the  surface  of 
the  water.  On  this  was  placed  a  caisson  of  iron  large  enough  to  contain  the 
pier,  and  with  sides  and  ends  high  enough  to  reach  to  the  level  of  high 
water  after  the  caisson  is  landed  on  the  bottom.  The  caisson  was  then  made 
water-tight.  The  bottom  was  then  floored  over  with  masonry  and  stone, 
and  laid  in  mortar  up  the  sides  of  the  caisson  to  the  top,  thus  constituting 
a  stone  caisson  inside  of  an  iron  one.  This  was  secured  to  the  guide-piles, 
and  the  masonry  of  the  pier  proper  was  laid  up,  the  caisson  sinking  as  the 
weight  of  masonry  inside  increased,  until  it  finally  settled  upon  the  bottom 
which  had  been  prepared  for  it,  as  already  described.  At  some  of  the 
piers  (figs.  5,  6)  screw-rods  were  used  to  suspend  the  pier  and  gearing 


ill 


Fig.  5. 


Fig.  6. 


attached,  governed  by  one  man,  who  at  pleasure  could  raise  or  lower  with- 
out assistance  the  whole  pier.  The  rock  was  reached,  and  the  masonry 
founded  upon  it  at  five  of  the  piers.  At  the  other  four,  after  the  dredging 
had  been  finished,  and  the  curb  was  sunk  to  the  required  depth,  wooden 


.     ENGINEERING   DRAWING.  421 

piles  were  driven,  and  cut  off  under  water  by  machinery  just  above  the 
ground,  and  the  platform,  with  its  incumbent  pier,  lowered  upon  them. 

Plate  CX.  is  a  transverse  section  of  the  river-wall  Thames  embank- 
ment, Middlesex  side.  It  may  be  said  to  be  a  wall  of  concrete,  &c.,  faced 
with  granite,  with  a  sewer  and  subway  within  the  same,  -both  enclosed  by 
brick-work.  In  the  drawings  the  different  material  is  represented  by  dif- 
ferent shadings;  and  letters  g  granite,  bb  brickwork,  cc  concrete.  To 
make  a  drawing  effective  it  is  often  expressed  by  tints  of  col- 
ors  suited  to  the  material,  and  the  concrete  with  dots,  thus — 

This  plate  and  the  following  are  not  only  to  show  the  im- 
portance of  the  work,  and  how  it  has  been  treated,  but  to  call 
attention  to  the  extensive  use  of  concrete,  and  show  what  dependence  is 
here  put  on  a  material  that  is  very  little  used  in  such  situations  in  this  coun- 
try. Extracts  are  given  from  the  specifications,  to  explain  the  construction : 

"  The  embankment-wall  is  to  be  formed  within  iron  caissons  or  coffer-dams,  as  the 
engineer  may  direct.  Should  caissons  be  adopted,  the  backs  and  fronts  thereof  are  iu 
all  cases  to  be  left  in  the  ground,  up  to  a  level  of  8  ft.  below  datum.  Should  coffer-dams 
be  adopted,  the  whole  of  the  piles  are  to  be  cut  off  to  such  a  level  as  the  engineer  may 
point  out,  that  no  danger  may  arise  to  the  several  works  from  any  drawing  of  piles.  As 
soon  as  the  excavations  shall  have  been  made  to  the  requisite  depths,  and  the  works 
cleared  of  water,  the  trenches  shall  be  filled  up  with  concrete  to  a  level  of  12|  ft.  below 
datum,  and  a  bed  dressed  to  the  proper  slope  and  level  for  the  footings  of  the  brick- 
wall.  This  wall  shall  be  formed  thereon  (when  the  concrete  has  become  thoroughly 
bard  and  consolidated)  at  a  true  slope  in  sets-off,  as  shown  on  drawing.  The  brick- 
work generally  shall  be  laid  in  courses  at  right  angles  to  the  face  of  the  wall.  The 
low  level  sewer  is  to  be  formed  on  concrete  foundation  carried  down  as  shown.  The 
sewer  shall  be  7'  9"  in  the  clear  diameter  for  a  length  of  1,820  ft.,  and  8'  3"  in  diameter 
for  the  remainder  of  its  length,  the  whole  to  be  formed  in  brickwork  1  ft.  1J"  thick. 
The  subway  shall  be  formed  7'  6*  high  by  9  ft.  wide  in  the  clear,  generally ;  the  side- 
walls  to  be  18",  the  arch  1'  1|"  thick.  The  subway-sewer  and  river-wall  shall  be  tied 
into  each  other,  at  intervals  of  6  ft.,  by  cross  or  counterfort  walls  18*  thick,  extending 
from  the  brickwork  of  the  wall  to  a  vertical  line  9"  beyond  the  side  of  the  sewer  farthest 
from  the  said  wall,  and  from  footings  9  ft.  below  datum,  which  are  to  be  bedded  on  a 
concrete  foundation  12"  thick,  up  to  the  under  side  of  the  subway.  The  upper  arch  of 
the  subway,  and  all  other  similar  arches,  shall  be  coated  on  their  outside  circumference 
with  a  layer  of  Claridge's  Patent  Seyssel  Asphalte,  1"  thick,  laid  on  hot,  and  returned 
up  all  spandrel  walls  rising  above  the  arch  to  a  height  of  9".  The  river-wall  shall  be 
faced  with  granite,  generally  to  a  level  of  8  ft.  below  datum,  and  shall  be  surmounted 
with  a  moulded  parapet  of  solid  granite ;  the  stones  to  be  laid  in  courses,  as  far  as  prac- 
ticable to  the  dimensions,  in  alternate  headers  and  stretchers,  and  as  shown  in  the  draw- 
ings. There  are  to  be  *  lamp-pedestals,  also  cast-iron  lions'  heads,  one  to  be  fixed  to 
each  pedestal,  with  cast  and  hammered  gun-metal  mooring-ring,  18"  inner  diameter,  and 
secured  to  the  counterfort  behind  the  subway  by  a  cast  iron  washer-plate  and  a  flat 


422  ENGINEERING   DRAWING. 

wrought-iron  mooring-bar.  The  lions'  heads  shall  be  thickly  coated  with  copper  on  the 
face  and  bed  by  the  galvanic  process,  or  cast  in  bronze,  as  may  be  directed  by  the  engineer. 

u  The  sand  for  the  various  works  is  to  be  very  clean,  sharp,  washed  river-sand.  The 
ballast  for  the  whole  of  the  concrete  is  to  be  very  clean,  and  the  material  is  to  be  well 
washed  before  it  is  delivered  on  the  works.  The  whole  of  the  cement  for  these  works 
to  be  Portland  cement  of  the  best  quality,  ground  extremely  fine,  and  weighing  not  less 
than  112  Ibs.  to  the  imperial  (striked)  bushel.  The  whole  of  the  concrete  to  be  used 
throughout  the  works  to  be  formed  of  1  measure  of  cement  to  6  equal  measures  of  ballast ; 
for  all  foundation-works,  and  for  all  other  work,  1  of  cement  to  8  of  ballast — the  whole 
to  be  thoroughly  mixed  by  machines.  Should  any  te  allowed  to  be  mixed  by  hand,  no 
greater  proportion  than  6  of  ballast  to  1  of  cement  will  be  permitted  to  be  used.  The 
lime  that  may  be  required  to  be  used  in  any  part  of  the  works  shall  be  of  a  quality 
equal  to  the  best  Aberthaw,  brought  in  lumps  fresh  from  the  kiln,  and  ground  upon  the 
works  in  mills  under  edge-runners.  -  The  mortar  to  be  mixed  in  proportions  of  1  of  lime 
to  2  of  sand,  well  tempered,  and  ground  in  similar  mills  for  25  minutes  at  least,  adding 
the  necessary  quantity  of  water  from  time  to  time.  The  bricks  throughout  to  be  of  the 
best  and  hardest  quality  ;  all  bats  brought  on  the  works  will  be  at  once  rejected.  The 
brickwork  to  be  executed  in  the  most  workmanlike  manner,  each  course  flushed  in, 
grouted,  and  finished  solid  ;  the  courses  laid  truly  parallel,  evenly,  and  uniformly  to  the 
curvature  of  the  works ;  and  centres  in  neat,  close,  and  regular  joints,  kept  straight  or 
regularly  curved,  as  the  case  may  be ;  and  the  joints  struck  neatly  and  flush  with  the 
face  of  the  work.  The  work  to  be  formed  in  rings,  Old  English  or  other  bond,  or 
herring-bone  courses,  when  and  as  may  be  directed  ;  the  brickwork  throughout  to  be 
set  in  Portland  cement,  mixed  with  not  more  than  an  equal  measure  of  sand,  and  which 
shall  not  be  remixed  or  used  after  it  has  become  set. ,  Where,  however,  the  works  are 
affected  by  water,  or  where  pointing  is  required,  neat  cement  is  to  be  used. 

"  All  granite  to  be  used  in  the  various  works,  especially  when  in  exposed  faces,  to  be 
of  uniform  color,  and  free  from  stains,  flaws  or  other  imperfections,  and  all  to  be  sound 
and  fully  equal  to  the  samples  deposited  at  the  office  of  the  Board.  In  the  embankment- 
wall  the  stones  are  to  be  fine  dressed,  and  axed  on  the  face  to  the  true  curved  batter,  so  as 
to  present  a  fair  and  perfect  surface ;  the  beds  and  joints  to  be  full  and  square  for  the 
whole  depth,  so  that,  when  set,  the  work  may  be  close  and  solid  throughout ;  and  no 
joint  to  exceed  ^"  in  thickness.  The  whole  of  the  stones  above  the  given  level  (111  ft. 
above  datum)  to  be  dowelled  together  in  bed  and  joints  with  slate-dowels,  not  less  than 
5  for  every  foot  run  of  wall;  each  2J*  square  at  least,  let  fully  2^"  into  each  stone,  very 
accurately  fitted,  and  run  in  with  neat  cement ;  the  stones  to  be  bedded  and  jointed  in 
cement,  and  the  joints  struck  with  neat  cement. 

"  The  cast  iron  shall  be  from  the  best  gray  cold-blast  pig-iron,  free  from  cinder, 
mixed  with  £  of  best  scrap-iron,  and  smelted  in  the  cupola.  Sample-bars  shall  be 
cast  for  the  purpose  of  testing ;  each  bar  to  be  2"  deep  by  V  wide,  placed  on  supports, 
with  a  clear  bearing  of  3  ft.,  and  to  bear  without  breaking  a  weight  of  3,200  Ibs.  when 
placed  upon  the  centre  between  the  supports.  The  wrought  iron  to  be  of  the  best 
quality  of  Staffordshire,  and  to  sustain  without  breaking  a  load  of  22  tons  per  square 
inch.  The  castings  to  be  clean  and  sound,  free  from  porous  places,  sand(  and  air-holes ; 
and  these  and  the  wrought-iron  work  to  be  free  from  all  other  imperfections.  The  whole 
to  be  delivered  on  the  works  perfectly  free  from  paint  or  other  coatings." 


ENGINEERING   DBA  WING.  423 

Plate  CXI.,  is  an  isometrical  view  of  the  overflow  and  outlet  of  the  Vic- 
toria and  Eegent  Street  sewers  in  the  Thames  embankment.  S  is  the 
main  sewer,  and  "W  the  subway  shown  in  plate  CX. ;  s  s  s  the  street-sewers, 
discharging  into  the  overflow  basin  O ;  w  w  the  weirs  over  which  the  water 
is  discharged  into  the  weir-chamber  c  c ;  p  is  the  penstock-chamber,  which 
is  but  a  continuation  of  the  weir-chamber.  It  has  been  attempted  in  the 
drawing,. by  breaks,  to  explain,  as  far  as  possible,  the  whole  construction; 
its  purpose  and  mode  of  action  are  perfectly  simple.  Whenever,  from 
storms,  the  discharge  from  the  street-sewers  (s  s  s)  is  greater  than  can 
be  carried  off  by  the  main  sewer  (S),  the  water  rises  in  the  overflow- 
chamber  (0),  passes  over  the  weirs  (w  w)  down  into  the  weir-chamber  (c), 
then  into  the  penstock-chamber,  and  through  the  flap-gates  (g)  into  the  river. 

Brief  extracts  from  the  specifications  will  be  found  useful  for  further 
explanation : 

"  The  foundation  to  be  of  concrete,  not  less  than  2  ft.  in  thickness ;  upon  this  brick- 
work shall  be  built  for  the  flooring  of  the  chambers,  and  for  the  side-end  and  weir-walls. 
TLie  weir-chamber  shall  be  divided  in  the  direction  of  its  length,  by  a  brick-wall, 
into  two  rectangular  overflow-channels,  covered  with  cast-iron  plates,  6'  8J"  long,  3' 
wide  by  Y  general  thickness,  with  strong  ribs  and  flanges  on  the  under  side,  properly 
bolted  together  and  jointed  with  iron  cement,  and  bolted  down  to  stones  which 
are  to  be  built  into  the  under  side  of  the  brickwork  of  the  basement-chamber.  Arches 
on  either  side,  running  parallel  thereto,  and  communicating  with  this  chamber  and 
with  the  weirs  which  are  to  be  formed,  upon  which  weir-walls,  divided  so  as  to 
correspond  with  these  arches,  are  to  be  built  in  brickwork,  capped  with  granite 
blocks,  4'  long,  2'  deep,  and  2'  3"  in  the  bed.  The  floor  of  the  penstock-chamber  to 
be  formed  with  York  landings,  6"  thick,  having  a  fall  of  3"  to  the  river.  The  outlets 
for  the  penstock-chamber  through  the  river-wall  shall  be  formed  by  an  arch-recess  in 
granite,  and  fixed  with  two  tidal  flaps,  well  hung,  and  firmly  secured  to  the  masonry 
by  strong  bolts  and  screws. 

"  The  subway  is  to  be  continued  over  the  low  level  sewer,  and  across  the  overflow- 
chamber,  by  cast-iron  plates,  curved  to  the  form  of  the  arch,  I"  general  thickness,  with 
strong  ribs  and  flanges  on  the  upper  side,  properly  bolted  together,  and  strongly  bolted 
down  to  the  brickwork ;  jointed  with  iron  cement,  and  covered  with  brickwork,  to 
form  the  floor  of  the  subway.  From  a  point  of  10'  3"  on  either  side  of  the  central  lon- 
gitudinal line  of  the  chamber,  where  the  sewer  and  subway  are  farthest  from  the  river- 
wall,  these  are  again  to  be  brought  into  their  general  position  by  two  curves,  each  not 
less  than  80  ft.  in  length. 

"  The  whole  of  the  cast  iron  shall  receive  one  coat  priming  of  red  lead  and  linseed 
oil,  and  three  coats  best  coal-tar,  before  fixing ;  and  the  accessible  surfaces  one  further 
coat  best  coal-tar,  when  fixed." 

Fig.  7  is  a  section  of  the  crib-pier  erected  on  the  west  bank  for 
the  Quarantine  establishment  for  the  port  of  New  York,  after  plans  by 


424 


ENGINEERING   DRAWING. 


Mr.  J.  "W.  Ritch.     The  structure  consists  of  an  outer  wall  of  crib-work, 
with  an  interior  filling  of  sand.     In  outline  it  is  like  a  bridge-pier,  228  ft. 


Fig.  7. 

wide  by  488  ft.  long,  measuring  between  the  extreme  points.     The  inter- 
ties  occur  at  intervals  of  6  ft.  spaces,  or  7  ft.  centres. 
Extracts  from  specifications : 

"  The  exterior  wall  to  be  built  in  blocks  up  to  low  water,  of  about  80  ft.  in  length, 
sunk  to  a  line,  and  to  be  filled  up  to  low  water  with  stone-filling,  before  the  remaining 
portion  is  commenced.  From  the  low  water  the  construction  of  the  exterior  wall  will  be 
continuous,  breaking  the  joints  of  the  logs  throughout  the  entire  length.  The  base  of 
the  blocks  will  be  formed  with  timbers,  14"  square ;  two  rows  on  the  outside,  held  to- 
gether with  interties  of  timber,  12"  square,  each  end  dovetailed  into  the  outside,  and 
shiplapped  to  the  other  timbers,  secured  at  each  end  and  intersection  with  iron  bolts,  1" 
square,  14"  long,  well  driven  home.  The  standards  are  to  be  10"  square  at  the  lower  end, 
and  long  enough  to  reach  above  low  water — all  to  be  placed  about  40  ft.  apart,  and  to  be 
secured  to  the  timbers  with  1"  square  bolts,  14"  long. 

"The  cribs  of  the  entire  exterior  wall,  from  the  foundation  to  the  top,  to  be  built  with 
sound  white  pine  or  spruce  timber,  12"  square,  laid  so  that  they  touch  each  other;  se- 
cured at  every  crossing  or  intersection,  and  in  the  centre  between  each  crossing,  with 
iron  bolts,  1"  square,  20"  long.  Eacb  end  of  each  floor-log  to  be  secured  to  the  interties 
with  iron  bolts,  1"  square,  12"  long.  Deck-timber  to  consist  of  one  under  tier  of  cross- 
timber,  and  one  upper  tier  of  cross-timber,  each  12"  x!2",  one  tier  of  ranging  timbers, 
each  14"  x  14".  The  cross-timbers  to  be  all  in  one  length ;  the  ranging  timbers  to  be  in 
lengths  of  not  less  than  46  ft. ;  joints  broken  over  the  logs  below.  The  cross-timbers  to 
be  dovetailed  at  the  ends,  and  shiplapped  at  intersections.  The  under  tier  of  timbers  to  be 


ENGINEERING   DRAWING.  425 

secured  to  the  logs  below,  the  ranging  timbers  to  the  under  tier,  and  the  upper  tier  to  the 
ranging  timbers,  as  follows,  at  each  end  and  every  crossing  with  an  iron  bolt,  1"  square, 
21"  long,  well  driven  home.  Also  4  stay-plates  to  each  row  of  cross-timbers,  each  plate 
to  be  |"x3i"  iron,  8  ft.  long,  secured  to  timber  and  logs  with  9  iron  spikes,  f  square, 
10"  long.  The  string-piece  to  be  12"  x  12",  secured  to  timbers  below,  every  7  ft.  with 
iron  bolts,  11"  square,  30"  long,  well  driven.  Tlie  entire  deck  to  be  covered  with  white 
pine  plank,  averaging  12"  wide,  4"  thick,  secured  by  £-"  iron  spikes,  11"  long.  The  entire 
exterior  to  be  close  fendered,  extending  from  the  deck-plank  to  low  water,  with  sawn 
white-oak  plank,  5"  thick,  and  not  over  12"  wide ;  each  plank  to  be  secured  with  7  iron 
bolts,  3"  square,  15"  long.  The  6  corners  of  this  fendering  to  have  each  3  iron  bands,  5 
ft.  long  on  each  limb,  3|"  x  1"  counter-sunk  holes  to  receive  5  iron  bolts,  J"  square,  15" 
long,  in  each  limb. 

"  Each  crib  to  be  tilled,  from  the  floor-logs  to  within  6"  of  the  deck-plank,  with  stone, 
granite,  gneiss,  or  trap-rock ;  none  of  the  stone  to  be  more  than  2  ft.  in  any  direction. 
The  entire  exterior  to  be  protected  with  stone,  in  large  pieces  .done  in  riprap,  extending 
from  the  base  horizontally  at  least  13  ft.,  and  perpendicularly  up  to  high  water,  each 
stone  to  be  not  less  than  1  ft.  thick,  5  ft.  long,  2  ft.  wide.  The  entire  space  inside  the 
wall  to  be  filled,  up  to  the  level  of  the  deck-plank,  with  sand  to  be  dredged  from  the 
shoal." 

Plate  CXII.  is  a  section  of  the  dam  across  the  Connecticut  Eiver,  at 
Holyoke,  Mass.  This  dam  is  1,017  ft.  long  between  abutments,  and 
averages  30  ft.  high  by  a  base  of  80  ft.  It  is  constructed  of  timber  crib- 
work,  loaded  in  with  stone  for  about  |  its  height.  The  foot  of  each  rafter 
is  bolted  to  the  ledge,  and  all  timbers  at  their  intersections  are  treenailed 
together  with  2"  white-oak  treenails.  The  inclined  plank-face  is  loaded 
with  gravel,  and  the  joint  at  the  ledge  covered  with  concrete.  The 
lower  or  base-tier  of  ranging  timbers  were  15"  x  15" ;  the  other  tim- 
bers, 12"  x  12".  The  rafters  are  placed  vertically  over  each  other,  in 
bents  of  6  ft.  between  centres.  The  planking  was  of  hemlock,  6"  thick, 
with  oak  cross-planking  at  crest  of  dam,  4"  thick  at  bottom  and  8"  at  top. 
The  crest  was  plated  with  iron,  \"  thick,  5  ft.  wide.  During  the  construction 
the  dam  was  planked  first  about  30  ft.  on  the  incline ;  a  space  was  then  left 
of  about  16  ft.  width  by  sufficient  length,  through  which  the  water  flowed ; 
and  the  balance  of  the  dam  was  then  completed.  A  plank-flap  was  then 
made  for  the  opening,  and  when  every  thing  was  ready,  it  was  shut  down, 
and  the  pond  filled.  The  dam  was  built  under  the  direction  of  the  late 
Mr.  John  Chase,  and  since  its  construction  the  greatest  depth  of  water 
passing  over  the  crest  during  a  freshet  was  12'  6". 

Fig.  8  is  a  section  of  part  of  the  dam  across  the  Merrimack  Eiver, 
at  Lowell,  built  under  the  direction  of  Mr.  Jas.  B.  Francis.  It  was  laid 
dry,  with  the  exception  of  the  upper  face  and  coping,  which  was  laid  full 
in  cement. 


426 


ENGINEERING   DRAWING. 


The  horizontal  joints  at  the  crest  were  run  in  with  sulphur.     The  coping- 
stones  were  dowelled  to  the  face  and  together,  and  clamped  to  an  inclined 


Fig.  8.— Scale :  {"  =  1  ft. 

stone  on  the  lower  slope,  the  end-joint  between  these  stones  was  broken 
by  making  every  alternate  lower  stone  longer,  and  the  upper  shorter,  than 
shown  in  the  drawings. 

The  Cohoes  Dam  (fig.  9)  was  built  under  my  direction,  directly  below  an 
old  dam  of  somewhat  similar  construction  to  that  of  Holyoke.  The  old 
dam  had  become  very  leaky  and  worn,  and  the  overfall  had  in  many 

places  cut  deep  into  the 
rock,  and  in  some  places 
within  the  line  of  the  dam. 
It  was  therefore  proposed 
to  make  the  new  dam,  as  a 
roll  to  the  old  one,  to  dis- 
charge the  water  as  far  from 
the  foot  of  the  dam  as  pos- 
sible, and  to  keep  the  old 
Fig.  9.-Scaie:  Ty  =  ift.  dam  for  the  protection  of 


ENGINEERING   DRAWING.  427 

the  new.  The  exterior  of  the  dam  was  of  rock-faced  ashlar ;  the  caps  were  in 
single  length  of  10  ft.,  and  none  less  than  15"  thick  and  2  ft.  wide ;  they  were 
dowelled  together  with  two  galvanized  wrought-iron  dowels  each.  The 
whole  work  was  laid  full  in  cement,  the  20"  wall  next  the  old  dam  being 
laid  distinct  without  bond  into  the  rest  of  the  work.  The  whole  was 
brought  up  to  the  outline,  to  receive  the  cap-stones,  which  were  bedded  in 
cement ;  the  top-joints  were  then  run  or  grouted  in  neat  cement,  to  within 
about  6"  of  the  top  of  the  stone,  which  was  afterward  run  in  with  sulphur. 
Entire  length  of  overfall,  1,443  ft. ;  average  depth  below  crest  of  dam,  12  ft. 

In  the  examples  of  dams  given,  the  foundations  of  all  have  been  upon 
ledge,  and  where  the  body  of  water  which  may  at  any  time  discharge 
over  the  dam  is  large  and  the  fall  high,  it  is  especially  desirable  to  secure  a 
location  where  the  overfall  can  be  upon  solid  rock.  If  there  be  ledge  at 
the  side  of  the  river,  and  none  can  be  found  in  the  channel,  it  is  often  bet- 
ter to  make  a  solid  dike  across  the  river  and  above  the  level  of  freshets,  and 
cut  the  overfall  out  of  the  bank.  "When  from  any  circumstances  the  dam 
can  have  only  an  earth  foundation,  an  artificial  apron,  or  platform  of  tim- 
ber or  rock,  is  to  be  made,  on  which  the  water  may  fall,  and  break  up  a 
high  fall  by  a  succession  of  steps.  In  some  cases,  a  roll  or  incline,  like 
that  given  in  the  new  Cohoes  dam,  is  extended  to  the  bed  of  the  stream, 
and  continued  by  an  apron.  The  water  thus  rolls  or  slides  down,  and 
takes  a  direction,  as  it  leaves  the  apron,  parallel  with  that  of  the  bed  of  the 
stream.  But  care  must  be  taken  to  protect  the  outer  extremity  of  the 
apron  by  sheet-piling  and  heavy  paving,  as  the  current,  by  its  velocity, 
takes  with  it  gravel  and  all  small  rocks,  and  undermines  the  apron. 

Fig.  10  is  a  section  of  the  dam  across  the  Croton  Eiver,  constructed 
under  the  direction  of  Mr.  John  B.  Jervis,  for  the  supply  of  the  aqueduct 
for  the  city  of  New  York.  This  dam  was  built  on  an  earth  foundation, 
with  curved  roll  in  cut  stone,  extended  by  a  timber-apron  some  50  ft.,  sup- 
ported by  strong  crib-work.  Originally  there  was  a  secondary  dam  still 
farther  down,  to  throw  back-water  on  this  apron.  In  the  erection  of  this 
dam,  excavation  was  made  of  all  loose  material;  the  cribs  C  and  D 
were  built  up,  and  the  tops,  were  planked  ;  on  this  planking  were  carried 
up  the  cribs  F  and  G.  Between  these  piers  the  space  E,  as  well  as  e  be- 
low and  on  the  cribs,  was  filled  in  with  concrete ;  on  this  the  body  of  the 
dam  was  erected  in  stone-masonry,  laid  in  cement.  The  face-work  of 
granite  is  cut  to  admit  of  a  joint,  not  exceeding  T\  of  an  inch.  Above 
the  dam  is  an 'earth  embankment;  its  upper  part  protected  by  a  rubble- 
paving.  The  radius  of  the  granite-face  is  55  ft.,  and  the  dam  38  ft.  high 
from  level  of  apron  to  crest  of  dam. 


428  ENGINEERING   DRAWING. 


Fig.  10.- Scale:  sy  =  1  ft. 

Dams  are  constructed  to  pond  water  for  the  supply  of  cities  and  towns ; 
for  inland  navigation,  by  deepening  the  water  over  shoals,  and  the  feeding 
of  canals ;  and  for  power  in  its  application  to  mills  and  workshops.  To 
whatever  purpose  the  water  is  to  be  applied,  there  are  two  questions  to 
be  settled :  Whether  the  level  will  be  raised  high  enough  by  the  construc- 
tion ;  and  whether  the  flow  of  the  stream  be  sufficient  for  the  purpose  re- 
quired ;  and  further,  it  may  often  be  important  to  know  how  large  a  pond 
will  be  thus  formed,  how  ample  a  reservoir  for  unequal  flow,  or  intermit- 
tent use.  If  the  pond  be  small,  so  that  the  water  cannot  be  retained,  but 
the  supply  is  only  the  natural  run  of  the  stream  at  a  higher  level,  then  the 
minimum  flow  of  the  stream  is  the  measure  of  its  capacity. 

For  the  measuring  or  gauging  of  small  streams,  a  rectangular  notch,  or 
weir,  in  the  vertical  plane  side  of  a  dam  or  sheet-piling  across  the  stream, 
is  by  far  the  most  convenient  apparatus ;  for  the  calculation  of  the  dis- 
charge of  which  formulae  have  been  established  by  extensive  experiments, 
both  here  and  abroad  ;  none  of  which  have  been  of  more  practical  value 
than  those  conducted  by  Mr.  Jas.  B.  Francis,  and  published  by  him,  in 
"  Lowell  Hydraulic  Experiments."  The  formula  given  by  him,  which  is 
most  generally  applicable,  is 

Q  =  3.33  VTL  (L  —  0.2  H)  H 

— in  which  Q  is  the  discharge,  per  second,  in  cubic  ft. ;  L  the  length  of 
weir ;  and  H  the  height  of  water  above  the  edge  of  weir,  in  feet. 

Thus,  if  the  weir  were  8  ft.  long,  and  the  depth  over  the  weir  0.81 
ft.,  then  vH  =  0.9  L  —  0.2  H  =  8  —  0.162  =  7.838. 

Q  =  3.33  x  9  x- 7.838  x  0.81  =  19.03  cubic  ft. 


ENGINEERING  DRAWING.  429 

This  rule  is  based  on  experiments  in  which  the  contraction  was  com- 
plete both  at  sides  and  at  bottom,  the  up-stream  side  of  the  dam  was  a 
smooth  and  vertical  plane  for  the  distance  of  at  least  twice  the  depth 
of  H  below  the  edge  of  the  weir,  and  three  times  li  on  each  side ;  and 
also  the  edge  of  the  weir  was  not  so  thick,  either  at  bottom  or  sides, 
as  to  form  a  short  pipe  for  contact  with  the  water,  but  the  aperture  was 
effectively  as  if  cut  in  a  thin  plate.  The  water  should  also  have  a  clear 
overfall,  so  that  a  constant  film  of  air  may  be  beneath  the  whole  sheet 
of  water.  The  depth  H  is  to  be  measured  in  still  water,  say  thrice  the 
depth  H,  from  the  weir. 

0.2  H,  in  the  above  formula,  is  a  correction  of  the  length  L, 
for  the  effect  due  to  the  side-contraction ;  if  there  be  a  contraction  on 
one  side,  it  becomes  L  —  0.1  IT,  or  simply  L  when  there  is  no  side-con- 
traction. 

To  measure  large  streams  of  water,  there  are  various  methods  of  ap- 
proximation— by  determining  the  top  or  a  mean  velocity  in  some  portion 
of  their  channel,  when  the  sections  are  nearly  uniform ;  by  finding  the  fall 
in  a  determinate  distance,  and  the  average  cross-section.  But  a  course  of 
measurement  even  of  a  season  will  supply  only  a  means  of  guessing,  which 
may  as  accurately  be  determined  by  an  estimate  of  the  area  of  the  rain 
shed  supplying  the  stream,  and  taking  one-half  of  the  rainfall  as  the  dis- 
charge of  the  stream.  From  the  knowledge  of  the  country,  we  can  judge 
how  speedily  the  rainfall  may  be  discharged,  and  how  much  of  the  water 
.of  freshets  can  be  reserved  for  use  during  the  dry  season.  Streams 
which  flow  from  hilly  and  mountainous  regions  are  subject  to  brief  but 
heavy  freshets,  and  often  to  seasons  of  very  low  water  ;  but  streams  flow- 
ing through  plains,  and  connected  with  ponds  and  marshes,  are  slower  in 
discharging  the  rainfall  and  more  steady  in  their  supply.  To  overcome 
the  inconvenience  resulting  from  fluctuations  in  the  flow  of  streams,  storage- 
reservoirs  are  often  constructed  on  the  main  stream  or  its  brandies,  or  the 
natural  ponds  or  lakes  enlarged  and  controlled  by  gates.  In  view  of  the 
improvement  of  a  stream  by  the  construction  of  storage-reservoirs,  they  may 
be  fairly  considered  an  element  in  the  calculation  of  the  value  of  a  stream. 
Many  a  city  or  town  in  this  country  is  dependent  for  its  supply  on  the 
ponding  of  a  stream  which  in  summer  months  would  be  entirely  inade- 
quate. Mr.  Ellet  proposed  to  construct  reservoirs  on  the  tributaries  of  the 
Ohio,  for  its  supply  for  purposes  of  navigation  during  the  dry  season  ;  and 
there  was  completed  in  1866,  near  St.  Etienne,  in  France,  a  dam  across 
the  river  Furens,  which  ponds  all  the  water  of  the  greatest  rainfall,  even 
of  a  water-spout,  to  prevent  the  destruction  of  property  which  has  often 


430  ENGINEERING   DRAWING. 

resulted  from  severe  freshets,  and  to  reserve  the  water  for  the  useful  pur- 
poses of  mill-power  and  water-supply.  The  Croton  Aqueduct  Department 
are  now  constructing  on  one  of  the  tributaries  of  the  Croton  a  dam  to 
pond  water  for  the  increase  of  the  city-supply. 

Blodget,  in  his  tk  Climatology  of  the  United  States,"  says  that  "  in  this 
sense  of  permanence  as  a  physical  fact,  we  may  consider  the  quantity  of 
rain  for  a  year  as  a  surface-stratum,  on  the  Atlantic  slope  and  in  the  cen- 
tral States  of  3i  ft.,  which  may  be  diminished  to  half  this  quantity,  or  in- 
creased to  twice  as  great  a  depth  in  the  extreme  years.  But  with  such  an 
average  and  such  a  known  range,  we  may  deal  with  the  quantity  as  defi- 
nitely as  with  a  stream  of  which  we  know  the  mean  volume  and  the  ex- 
tremes to  which  it  is  liable,  and  for  many  departments  of  engineering 
these  climatological  measures  are  as  indispensable  as  those  of  tide  or  river 
hydrography." 

The  evaporation  from  a  reservoir-surface  at  Baltimore,  during  the  sum- 
mer months,  was  assumed  by  Colonel  Abert  to  be  double  the  quantity  of  rain- 
fall. Dr.  Holyoke  assigns  the  annual  quantity  evaporated  at  Salem,  Mass., 
to  be  56" ;  but  from  experiments  made  by  the  Croton  Aqueduct  Depart- 
ment, in  1864,  of  the  evaporation  from  a  box  set  in  the  earth-bank,  and 
two  afloat  in  the  upper  reservoir,  the  quantity  was  found  to  be  severally 
37.12,  3T.53,  and  39.97  inches. 

Head-gates  are  constructions  necessary  to  control  the  flow  from  the 
river-pond  or  reservoir  into  the  canal  or  conduit  by  which  the  water  is  to 
be  conveyed  and  distributed  for  the  purposes  to  which  it  is  to  be  applied. 
The  top  of  the  works  should  therefore  be  entirely  above  the  level  of  the 
highest  freshets,  that  no  water  may  pass,  except  through  the  gates ;  and  it 
is  better  that  the  opening  of  the  gates  should  be  entirely  below  the  level 
of  the  top  of  the  dam,  to  prevent  as  much  as  possible  the  passage  of  drift 
or  ice,  which  are  often  excluded  by  booms  and  racks  placed  outside  the 
gates. 

Plates  CXIII.  and  CXIY.  are  drawings,  in  plan  and  detail,  of  the 
head-gates,  and  the  machinery  for  hoisting  them,  at  the  Cohoes  Company's 
dam. 

It  will  be  seen,  by  reference  to  the  plan,  that  there  are  10  gates.  The 
dimensions  of  4  are  8'  x  6'  6" ;  and  6,  8'  x  9',  in  the  clear — all  of  which  can 
be  hoisted  by  machinery  connected  with  a  turbine-wheel  at  a,  or  separately 
by  hand.  At  b  there  is  an  overfall,  at  the  same  height  as  the  dam,  over 
which  any  drift  that  is  brought  against  the  gate-house  is  carried.  At  c, 
there  is  a  similar  overfall  within  the  gates,  and  another  at  d,  by  which  any 
sudden  rise  of  the  level  of  the  canal  is  prevented.  At  e,  there  is  a  gate 


ENGINEERING   DRAWING.  431 

for  drawing  down  the  pond,  and  another  at  f,  for  drawing  off  by  the 
canal,  both  raised  and  lowered  like  the  head-gates. 

The  head-gates  are  of  solid  timber  bolted  together,  moving  in  cast-iron 
guides  set  in  grooves  in  the  stone;  in  front  of  these  grooves  there  is  an- 
other set  of  grooves  (g  g),  which  are  intended  for  slip-planks  or  gates,  to 
be  put  in  whenever  it  is  necessary  to  shut  off  the  water  from  the  gates 
themselves  in  case  of  repairs.  Hoisting  apparatus. — To  each  gate  there 
are  strongly  bolted  two  cast-iron  racks,  geared  into  two  pinions  on  a  shaft 
extending  across  the  gate-space,  and  suppor!ed  on  cast-iron  standards  on 
the  piers.  At  one  extremity  of  this  shaft,  there  is  a  worm-wheel,  driven 
by  a  worm  or  screw  on  a  shaft  perpendicular  to  the  pinion-shaft.  The 
worm-shaft  can  be  driven  either  by  a  hand-wheel  at  one  end,  or  by  the 
friction-bevel  at  the  other.  The  friction-bevel  can  be  driven  in  either  direc- 
tion by  being  brought  in  contact  with  one  or  other  of  the  friction-bevels  on 
a  shaft  extending  the  whole  length  of  the  gate-house,  and  in  gear  directly 
with  the  small  turbine  at  a.  The  small  turbine  draws  its  supply  through 
a  pipe,  built  in  the  walls,  and  opening  into  the  space  between  the  gates  and 
the  slip-plank  groove. 

The  sections  of  canals  depend  upon  the  purposes  to  which  they  are 
to  be  applied,  whether  for  navigation  or  for  power:  if  for  navigation, 
reference  must  be  had  to  the  class  of  boats  for  which  they. are  intended; 
if  for  power,  to  the  quantity  of  water  to  be  supplied,  and  sundry  pre- 
cautions of  construction. 

Fig.  11  is  a  section  of  the  Erie  Canal :  width  at  water-line,  70  ft. ;  at 
bottom,  28  ft. ;  depth  of  water,  7  ft. ;  width  of  tow-path,  14  ft.  It  will  be 
observed  that  the  slopes  are  gravelled  and  paved,  and  that  the  edge  of  the 
tow-path  is  paved  with  cobble-paving,  and  the  path  gravelled.  The  smaller 


Fig.  11. 

canals  of  this  State  and  of  Pennsylvania  are  generally  40  ft.  wide  at  water- 
line,  and  4  ft.  deep  ;  the  Delaware  and  Karitan,  75'  x  7' ;  the  Chesapeake 
and  Delaware,  66'  x  10' ;  the  ship-canals  of  Canada,  10  ft.  deep  and  from 
70  to  190  ft.  wide. 

The  dimensions  for  canals  for  the  supply  of  mills  depend— first,  on  the 
quantity  of  water  to  be  delivered.  Their  area  of  cross-section  should  be 
such,  that  the  average  velocity  of  flow  should  not  exceed  2  ft.  per  second, 
and  'in  northern  climates  less  velocity  than  this  would  be  still  better ;  it 


432 


ENGINEERING   DRAWING. 


should  always  be  such,  that  during  the  winter  the  canals  may  be  frozen 
over,  and  remain  so,  to  prevent  the  obstruction  from  drift  and  anchor-ice 
in  the  water-wheels.  The  usual  depths  of  the  larger  canals  are  from  10  to 
15  ft. ;  with  such  depths,  the  cover  of  ice  which  reduces  the  section  by  the 
amount  of  its  thickness,  does  not  materially  increase  the  velocity  of  flow, 
nor  diminish,  consequently,  very  perceptibly  the  available  head. 

Fig.  12  is  a  section  of  the  Northern  Canal,  at  Lowell,  Mass.,  which 
may  be  considered  a  model  for  large  works.  The  width  at  water-line  is 
103  ft,,  and  the  depth  16' ;  and  is  intended  for  an  average  flow  of  2,700 
cubic  ft.  per  sec. ;  the  fall  in  the  whole  length  of  4,300  ft.  is  between 


Fig.  13. 

2"  and  3" ;  when  covered  by  ice,  about  4".  The  sides  are  walled  in  dry 
rubble,  and  coped  by  split  granite.  It  will  be  observed  that  the  portion 
above,  and  about  3  ft.  below,  the  water-line,  or  between  the  limits  of  ex- 
treme fluctuations  of  level,  is  laid  plumb,  that  the  ice  may  have  as  free  a 
movement  as  possible  vertically. 

Fig.  13  is  a  section,  on  a  scale  of  |"  =  1  ft.,  of  the  river-wall  of  this 


Fig.  13. 


ENGINEERING   DRAWING. 


433 


same  canal,  where  the  canal  passes  out  into  and  occupies  a  portion  of  the 
river-channel,  and  the  depth  of  water  in  the  canal  is  greater  than  in  above 
section.  The  main  wall  is  in  dry  masonry,  faced  on  river-side  with  rough- 
faced  ashlar,  pointed  beds  and  end-joints.  The  inside  lining  is  of  two 
courses  of  cement-wall,  the  dry  rubble  backing  being  first  laid,  then 
pointed  with  cement,  against  which  is  laid  the  first  cement  lining,  which  is 
plastered  on  the  inside,  and  the  interior  wall  is  then  laid ;  the  granite 
inside  wall,  above  lining,  is  also  laid  in  cement. 


Fig.  15.— Scale:  ry  =  1ft. 

Locks  of  Canals. — Figs.  14  and  15  are  portions  of  plan  and  vertical 
section  of  locks,  taken  from  the  general  plans  for  timber  locks  on  the 
Chemung  Canal.  They  represent  the  half  of  upper  gates. 

28 


434 


ENGINEERING   DRAWING. 


Fig.  16  is  a  section  of  one 
side  of  the  lock  of  the  same. 

Fig.  17  is  the  plan  of  a  por- 
tion of  one  of  the  enlarged 
locks  of  the  Erie  Canal,  show- 
ing one  of  the  upper  gates  and 
the  side-walls. 

Fig.  18  is  a  cross-section  of 
one  of  the  same  locks,  showing 
the  culvert  in  the  centre  between 

the  locks,  used  for  the  supply  of  Fig.  ie. 

the  waste  of  the  lower  level,  and  to  preserve  the  proper  height  oi  water  in 
this  level. 


Fig.  18.— Scale :  iV  =  1  ft. 


ENGINEERING   DRAWING. 


435 


Fig.  19  is  a  drawing,  in  outline,  of  the  hollow  quoin  of  the  lock-gate 
on  a  scale  of  ft  fall  size  (Chemung  Canal). 


Fig.  80.-A  full  Bize. 

Fig.  20  is  a  plan  and  elevation  of  pintal  for  heel-post  of  lock,  with  a 
section  of  the  bottom  of  the  post.  The  pintal  is  imbedded  in  bottom 
timber  or  stone,  as  the  case  may  be. 

Fig.  21  is  a  plan  and  elevation  of  the  strap  for  the  upper  part  of  heel- 
post. 


Extracts  from  lock  specifications  (New  York  State  Canals,  1854) : 

"  I^ocks  to  be  composed  of  hydraulic  stone  masonry,  placed  on  a  foundation  of  tim- 
ber and  plank.  The  chamber  to  be  18'  wide  at  the  surface  of  the  water  in  the  lower 
level,  and  110'  long  between  the  upper  and  lower  gate-quoins.  The  side-walls  to  extend 
21'  above  the  upper  gate-quoins,  and  14'  below  lower  gate-quoins.  If  the  bottom  is  of 
earth,  and  not  sufficient  to  support  the  foundation,  then  bearing  piles  of  hard  wood,  not 
less  than  10"  diameter  at  small  end,  shall  be  driven,  to  support  the  foundation.  There 
shall  be  4  rows  of  piles  under  each  main  wall,  and  1  row  in  centre  of  lock;  the  piles 
shall  be  driven  in  rows,  at  3' from  centre  to  centre.  The  piles  to  support  the  wing  and 
breast-walls  and  wing  buttresses,  and  also  under  the  mitre-sills,  to  be  driven  in  rows  to 
conform  to  the  form  and  shape  of  the  same.  The  heads  of  the  piles  to  be  cut  off  smooth 
and  level,  to  receive  the  foundation  timbers.  The  foundation  timbers  to  be  12*xl2", 
and  of  such  lengths  as  will  exttnd  from  and  cover  the  outside  piles,  and  to  be  trcenailed 
with  a  2"  white-oak  or  white-elm  treenail.  24"  long,  to  each  pile. 


436  ENGINEERING   DRAWING. 

"  If  the  bottom  is  of  earth  sufficiently  compact  and  firm  to  support  the  foundation 
without  bearing  piles,  then  the  foundation  shall  be  composed  of  timber,  12*  thick  and  not 
less  than  10"  wide,  counterhewed  on  upper  side,  timbers  to  average  12*  wide,  to  be 
placed  at  uniform  distance,  according  to  their  width,  so  as  to  occupy  or  cover  at  least 
^  of  the  area  of  the  foundation,  and  under  the  lower  mitre-sill  to  be  placed  side  by  side : 
in  all  cases  to  be  of  sufficient  length  to  extend  across  the  lock  to  the  back  line  of  the 
centre  buttresses,  and  at  the  head  and  foot  to  the  rear  or  back  line  of  wing  walls.  The 
timber  under  the  lower  mitre-sill  to  be  of  white  oak,  white  elm,  or  red 'beech,  the  other 
foundation  and  apron  timber  to  be  of  hemlock.  The  foundation  to  be  extended  3'  above 
the  face  of  the  main  wall  at  the  head  of  the  lock,  and  at  the  foot  from  25'  to  30'  below 
the  exterior  wing— that  portion  of  the  spaces  between  the  timbers  in  all  cases  to  be 
filled  with  clean  coarse  gravel,  well  rammed  in,  or  concrete.  In  cases  where  rock  com- 
poses the  bottom  of  the  lock,  the  foundation  timbers,  if  required,  shall  be  10*  thick  un- 
der the  lower  mitre-sill,  and  8"  thick  at  other  places.  Where  the  rock  is  of  such  a  char- 
acter that  timber  is  not  required  for  the  foundation,  the  same  shall  be  excavated  smooth 
and  level,  and  the  first  course  of  stone  well  fitted  to  the  rock. 

"  Sheet-piling. — In  all  cases  where  rock  does  not  occur,  there  shall  be  a  course  at 
the  head  of  the  foundation,  under  each  mitre-sill,  and  at  the  lower  end  of  the  wings, 
and  at  the  lower  end  of  the  apron,  to  be  from  4'  to  6'  deep,  as  may  be  required — in  each 
to  extend  across  the  whole  foundation.  The  sheet-piling  to  be  of  2"  hemlock  plank,  lined 
with  1"  pine  boards.  Ditches  are  to  be  excavated  to  receive  the  sheet-piling,  which  are 
to  be  placed  edge  to  edge,  and  the  top  well  secured  to  the  foundation  timber ;  the  spaces 
to  be  filled  up  with  fine  hard  gravel,  well  puddled  in,  or  with  concrete. 

"Flooring. — A  course  of  2i*  pine  or  hemlock  plank  to  be  laid  over  the  whole  of  the 
foundation  timbers,  except  a  space,  8'  wide,  under  the  face  line  of  each  wall,  to  be  2£" 
white  oak:  the  whole  to  be  well  jointed,  and  every  plank  to  be  treenailed  with  2  white- 
oak  treenails  at  each  end,  and  at  every  3'  in  length,  to  enter  the  timber  at  least  5",  or 
with  wrought-iron  spikes,  treenails  to  fill  1£"  bore.  Platform  for  the  upper  mitre-sill 
to  be  5'  10"  wide,  and  6'  high  above  foundation,  and  to  extend  across  from  side-wall  to 
side-wall,  to  be  composed  of  masonry,  coped  with  white-oak  timbers,  which  are  to  extend 
6"  into  each  side-wall.  The  timbers  to  be  12"  deep  and  14"  wide,  covered  with  2  courses 
of  1£"  white-oak  plank.  Mitre-sills  to  be  of  best  white-oak  timber,  9"  thick,  to  be  well 
jointed,  and  bolted  to  the  foundation  or  platform  timbers,  as  the  case  may  be,  with  bolts 
of  iron,  20"  long,  1"  x  1",  well  ragged  and  headed,  8  bolts  to  each  side. 

"Masonry. — The  main  walls,  for  21'  6"  in  length,  from  wing-buttresses  at  the  head, 
and  32'  at  lower  end,  to  be  9'  8^"  thick,  including  recesses,  and  for  the  intermediate 
space,  7"  8|"  thick,  with  3  buttresses  projecting  back  2|',  and  9'  long  at  equal  distances 
apart.  The  quoin-stones,  in  which  the  heel-post  is  to  turn,  shall  not  be  less  than  4'  6"  in 
length  in  line  of  the  chamber,  to  be  alternately  header  and  stretcher.  The  recesses  for 
the  gates  to  be  20"  wide  at  top  of  wall,  12'  long,  with  sub-recesses,  9"  wide,  6'  high,  10' 
long,  for  the  valve-gates.  Breast-wall  to  commence  5'  below  upper  end  of  foundation, 
5'  wide,  8'  high,  finished  with  a  coping  of  cut  stone.  The  interior  wing-walls,  and  ex- 
terior wing  from  main  walls  to  the  termination  of  first  curve,  to  be  V  6"  thick,  and  the 
running  curve  of  exterior  wing  to  be  6'  thick  on  the  foundation. 

"  Culvert  "between  Locks. — In  such  cases  as  may  be  required,  a  culvert  shall  be  con- 
structed, to  pass  the  water  from  the  upper  to  the  lower  level,  as  follows :  A  foundation 


ENGINEERING    DRAWING.  437 

of  suitable  timber  and  plank,  as  for  lock-walls,  and  covering  all  the  space  between  the  lock- 
'  foundations,  shall  be  put  down.  Three  apertures  for  the  sluice-way  shall  be  made  in  the 
head-wall  with  cut-stone  jambs,  grooves  to  be  cut  in  the  jambs  for  the  sluice-gates,  and 
the  coping  to  form  a  recess,  corresponding  with  the  grooves  in  the  jambs;  grooves  to  be 
cut  on  the  top  and  bottom  coping,  1"  deep,  to  secure  the  jambs.  The  bottom  of  the  aper- 
ture to  be  of  cut  stone,  with  lower  corner  bevelled  off,  over  which  the  water  will  fall  into 
the  well,  the  bottom  of  which  shall  be  covered  with  a  sheeting  of  cut  stone,  6"  thick. 
The  apertures  to  be  3'  6"  deep,  placed  immediately  below  the  coping-stone,  and  4'  long. 
Suitable  gates  of  plank,  for  regulating  the  water  in  passing  the  sluice,  to  be  prepared;  the 
well  to  commence  on  the  foundation,  to  be  made  of  substantial  hydraulic  masonry. 

"  Second  flooring  of  seasoned  2"  first-quality  white-pine  plank,  to  be  well  jointed,  and 
laid  on  the  foundation  between  the  walls,  from  the  breast-wall  to  lower  end  of  main  wall, 
and  also  on  the  floor  of  the  well,  to  be  close  and  firmly  jointed  to  mitre-sills  and  walls,  so 
as  to  make  a  water-tight  flooring.  The  plank  to  butt,  or  the  end-joints  to  come  to  the 
centre  of  a  foundation  timber,  and  each  plank  to  be  treenailed  with  2  treenails  at  end  and 
2  at  every  3'  intermediate  :  treenails  10"  long,  to  fill  !]•"  bore. 

"  Gates. — The  framing  to  be  made  of  best-quality  white-oak  timber ;  the  cross-bar  to 
he  framed  into  heel  and  toe  posts  with  double  tenons,  each  tenon  to  be  V"  long,  and  thick- 
ness equal  to  the  thickness  of  the  bar,  and  secured  with  wrought-iron  Ts,  well  bolted. 
The  heel  and  toe  posts  to  be  framed  to  the  balance-beam  by  double  tenons,  and  secured 
by  a  wrought-iron  strap  and  balance-rod,  from  the  top  of  the  beam  to  the  under  side  of 
the  upper  bar.  The  lower  ends  of  the  heel-posts  to  be  banded  with  wrought-iron  bars ; 
the  collar  and  other  hangings  to  be  of  wrought  iron,  secured  together  with  a  double  nut  and 
screw,  and  to  the  coping  by  bedding  the  depth  of  the  iron  in,  and  by  screw-bolts  fastened 
with  sulphur  and  sand-cement.  The  pivots  and  sockets  which  support  the  heel-posts  to 
be  of  best  cast  iron;  a  chilled  cast-iron  elliptical  ball,  2£"  horizontal,  and  1"  vertical 
diameter,  to  be  placed  on  the  pivot  and  in  the  socket  of  each  heel-post,  to  facilitate  the 
movement  of  the  gate.  The  gates  to  be  planked  with  seasoned  first-quality  2"  white-pine 
plank,  jointed,  grooved,  and  tongued — tongues  of  white  oak — the  plank  to  be  secured  by 
6"  pressed  spike.  On  the  chamber-side  of  the  gates,  fenders  of  white-oak  plank,  to  be 
put  on  with  pressed  spike." 


Water,  ponded  by  dams,  and  conveyed  by  canals  for  use  as  mill-p6*wer, 
is  carried  within  the  workshops  or  manufactories,  to  be  applied  on  water- 
wheels,  by  some  covered  channels.  These  channels,  although  of  various 
forms,  are  usually  designated  as  flumes.  The  common  form  of  a  flume  for 
the  conveyance  of  water  to  breast,  overshot,  or  undershot  wheels,  is  of  a 
rectangular  section,  framed  with  sills,  side-posts,  and  cap,  and,  if  large 
section  is  required,  intermediate  posts  are  set  in.  The  sills  are  set,  and 
earth  well  rammed  in  the  spaces  between  them ;  the  bottom  plank  is  then 
laid,  posts  and  cap  framed  with  tenon  and.  mortice,  set  and  pinned,  and 
the  plank  is  then  firmly  spiked  on  outside  of  posts  and  caps.  The  planks 
are  usually  nearly  green,  jointed,  and  brought  to  close  joints ;  the  size  of 
timbers  will  depend  on  the  depth  beneath  the  soil,  or  the  insistent  load. 


438  ENGINEERING   DRAWING. 

Within  the  mill,  and  just  above  the  wheel,  the  flume  is  framed  with- 
out a  cover,  and  the  posts  and  side-planks  are  brought  above  the  level 
of  the  water.  This  open  flume  is  termed  the  penstock,  especially  ne- 
cessary in  the  class  of  wheel  above  referred  to,  to  secure  the  full  head  of 
water. 

Many  flumes  are  made  of  a  circular  section,  pipes  of  iron,  or  wood. 
For  the  conveyance  of  water  to  turbine-wheels,  wrought-iron  pipes  are 
almost  invariably  used.  (Plates  XLII.  and  XLIII.)  Cast  iron  is  also 
sometimes  used,  with  flange,  or  hub  and  spigot-joints.  Plank-pipes  are 
also  used,  made  with  continuous  staves,  and  hooped  with  wrought  iron : 
these  constructions  are  much  cheaper,  and  serve  a  very  good  purpose. 
The  head-gates  of  flumes  are  placed  at  the  head  of  the  flumes,  in  a  recess 
back  from  the  face  of  the  canal,  with  racks  in  front  to  prevent  the  passage 
of  any  drift  that  might  obstruct  or  injure  the  wheel.  The  total  area  of 
passages  through  the  racks  should  liberally  exceed  the  area  of  cross-section 
of  the  flume,  not  only  on  account  of  the  extra  lateral  friction  of  the  rack- 
bars,  but  also  on  account  of  their  liability  to  become  obstructed.  Some- 
times two  sets  of  racks  are  placed  in  front  of  the  flumes,  especially  for 
turbines  and  reacting  wheels :  a  coarse  rack  with  wide  passages,  say  2" 
spaces  outside,  and  a  finer  one  inside,  say  of  £"  to  %"  spaces.  The  head- 
gates  to  the  flume,  directly  back  of  the  racks,  in  their  function  are  like  the 
head-gates  at  the  dam,  and  are  similar  in  construction — strong  plank  gates, 
moving  in  slides,  vertically  or  horizontally,  with  a  paddle-gate  in  them, 
to  fill  the  flume  when  empty,  so  that  the  gates  themselves  may  be  opened 
without  any  pressure  due  to  a  difference. of  head  outside  and  inside  of  the 
gates,  and  also  to  prevent  any  damage  to  the  flume  by  the  water-ram, 
which  might  result  from  a  too  sudden  filling  of  the  flume  by  the  opening 
of  a  large  gate  suddenly. 

Plate  CXY.  is  the  elevation  and  section  of  the  head-gates  manufactured 
at  Holyoke,  Mass.  G,  G  are  plank  gates,  sliding  laterally,  moved  by  two 
pinions,  working  into  racks  on  top  and  bottom  of  gates,  turned  by  a  hand- 
spike. P  is  the  paddle-gate ;  E,  the  rack ;  F,  the  flume,  or  plank-pipe ; 
A,  air-pipe,  for  the  escape  of  air  from  the  flume  while  being  filled. 

Conduits  for  the  conveyance  of  water  for  the  supply  of  cities  and  towns 
should  always  be  covered,  and  of  a  capacity  adapted  to  the  quantity  to  be 
delivered.  Capacity  is  determined  by  area  of  section,  descent  or  loss  of 
head,  and  length  and  directness  of  conduit. 

Fig.  22  is  a  cross-section  of  the  main  conduit  of  the  Nassau  Water- works 
for  the  supply  of  the  city  of  Brooklyn,  L.  I.  The  width  is  10'  at  the  springing 
of  the  arch ;  the  side-walls  3  ft.  in  height ;  versed  sine  of  invert,  8" ;  height  of 


ENGINEERING    DRAWING. 


439 


conduit  in  centre,  8'  8" ;  fall  or  inclination  of  bottom,  1  in  10,000  ;  full  capa- 
city, 47,000,000  K  Y.  galls,  in  24  hours.  In  preparation  of  the  foundations 
the  contract  specifications  required  a  bed 
of  concrete  to  be  first  laid,  15'  wide  ;  but, 
when  the  water  was  very  troublesome, 
it  was  found  necessary  to  lay  a  plat- 
form of  plank  for  the  concrete.  The 
side-walls  are  of  stone,  except  an  interior 
lining  of  4"  brickwork.  The  arch  is 
brick,  12",  and  the  invert  4"  thick.  The 
outside  of  arch,  as  it  was  finished,  and 
each  wall,  were  plastered  over  on  the  out- 
side with  a  thick  coat  of  cement-rnortar. 
The  concrete  was  formed  from  clean  bro-  Fig.  22.-scaie,  j"  =  i  ft. 

ken  stone,  broken  so  as  to  pass  through  a  2"  ring ;  2  to  2£  measures  of  broken 
stones  were  mixed  with  1  measure  cement-mortar.  The  centres  of  the  arch- 
ing were  not  allowed  to  be  struck  until  the  earth  had  been  well  packed  in 
behind  the  side-walls  and  half-way  up  the  arch.  In  both  cuttings  and  em- 
bankments- the  arch  was  covered  with  4  ft.  of  earth,  with  a  width  of  8  ft. 
at  top,  and  slopes  on  each  side  of  1£  to  1,  covered  with  soil  and  seeded  with 
grass. 

Fig.  23  represents  a  section  of  the  Croton  Aqueduct,  in  an  open  rock- 
cut.  The  width  at  spring  of  arch,  T ;  versed  sine  of  invert,  6" ;  height 
of  conduit,  8'  6"  ;  fall  or  inclination  of 
bottom,  about  1  in  5,000  ;  flow,  where 
there  is  5'  10"  in  depth  of  water  at 
centre,  60,000,000  K  Y.  galls,  per  24 
hours.  The  bottom  is  raised  wTith  con- 
crete to  the  proper  height  and  form 
for  the  inverted  arch,  of  a  single 
course  of  brick ;  the  side-walls  are  of 
stone,  laid  in  cement,  plastered,  and 
laced  with  a  single  course  of  brick ; 
the  arch  is  semicircular,  of  brick  two 

courses  thick,  with  spandrel  backing  nearly  to  the  level  of  the  crown,  and 
earth  filled  on  the  top.  In  earth-cuts  or  embankments,  side-walls  were 
constructed  of  stone,  in  cement;  and  in  embankments  the  whole  structure 
rested  on  dry  rubble-walls,  built  up  from  solid  earth-foundations. 

At  the  crossing  of  the  Harlem  Kiver,  as  the  bridge  was  depressed  be- 
low the  level  of  the  aqueduct,  the  water  was  conveyed  by  two  cast-iron 


440 


ENGINEERING   DRAWING. 


pipes,  a  a,  3'  in  diameter,  fig.  24 ;  but,  as  the  demand  for  water  increased  in 
the  city,  the  obstruction  caused  by  lack  of  capacity  in  these  pipes  has 
made  necessary  the  introduction  of  a  larger  pipe,  which  has  been  made  of 
wrought  iron,  %"  thick  and  T'  6£"  in  diameter ;  this  is  supported  by  cast- 


Fig.  24. 

iron  columns  which  admit  of  a  rocking  movement,  and  slip-joints  are  also 
made  in  the  pipe,  to  compensate  for  any  expansion  or  contraction  by 
change  of  temperature.  The  pipes  are  enclosed  in  a  long  chamber  or  pas- 
sage, extending  the  whole  length  of  the  bridge,  covered  by  a  brick  arch, 
laid  in  cement,  and  protected  by  asphalt,  with  a  brick  pavement  over  all. 
A  A  are  the  arch-stones  of  the  bridge,  and  C  C  coping  stones,  in  which 
are  inserted  the  posts  of  a  cast-iron  railing. 

In  large  works,  where  there  is  considerable  length  of  conduit,  receiving 
reservoirs,  within  or  near  the  limits  of  the  city,  are  necessary  as  a  precaution 
to  guard  against  accidents  which  might  happen  to  conduit  or  dam,  and  cut  off 
the  supply,  and  also  as  a  sort  of  balance  against  unequal  or  intermittent 
draught  among  the  consumers.  The  size  of  these  reservoirs  must  depend  on 
the  necessities  of  the  case,  and  on  the  facilities  for  construction.  The  capa- 
city of  the  Kidgewood  reservoir,  at  Brooklyn,  is  161,000,000  N.  Y.  gal- 
lons, when  full ;  of  the  new  Croton  reservoir,  about  1,000,000,000  gallons. 
Both  these  reservoirs  are  made  double,  that  is,  in  two  compartment;. 

Fig.  25  is  a  section  of  the  division-bank  of  the  new  Croton  reservoir. 
It  is  made  of  earth,  with  a  puddled  ditch  in  the  centre,  and  slopes  pro- 
tected by  rock-paving. 


ENGINEERING   DRAWING. 


441 


Fig.  25. 

A  few  extracts  from  the  specification  will  explain  the  general  con- 
struction of  the  reservoir : 

"  The  reservoir  will  be  formed  by  an  exterior  bank  forming  the  outer  sides  of  the 
basin.  There  will  be  a  division-bank,  dividing  the  reservoir  into  two  basins.  All  the 
banks  will  have  the  inner  and  outer  slopes  of  \\  base  to  1  perpendicular.  All  the  inner 
or  water-slopes  will  be  covered  with  8"  of  broken  stone,  on  which  will  be  placed  the 
stone  pavement,  H  ft.  thick.  The  outer  slopes  will  be  covered  with  soil,  1  ft.  thick. 
The  banks,  when  finished,  to  be  15  ft.  on  top,  exclusive  of  the  soil  on  the  outer  slope. 
The  top  of  the  outer  bank  to  be  4  ft.  above  water-line  ;  the  top  of  the  division-bank  to 
be  3  ft.  below  water-line.  In  the  centre  of  all  the  banks  a  puddle-bank  will  be  built,  ex- 
tending from  the  rock  to  the  paving  in  the  division-bank,  and  to  within  2  feet  of  the  top 
of  the  outer  bank.  It  will  be  6'  2"  wide  at  top  in  division-bank,  and  14'  wide  at  top  in 
exterior  bank,  and  16'  wide  at  a  plane  38'  below  top  of  exterior  bank.  In  the  middle 
of  the  division-bank  there  will  be  built  a  brick-wall,*  laid  in  cement-mortar,  4'  high,  20" 
wide,  the  top  of  the  wall  to  be  connected  with  the  bottom  of  the  stone  pavement;  8" 
thickness  of  concrete  is  to  be  laid  on  the  top  of  the  bank,  on  each  side  of,  and  connected 
with,  this  wall.  On  the  pavement  18"  thick  will  be  laid  in  concrete.  The  slope-wall  on 
each  side  of  the  division-bank,  10'  in  width,  to  be  laid  in  cement. 

"  Puddle-ditches  are  to  be  excavated  to  the  rock  under  the  centre  of  all  embank- 
ments where  the  rock  is  not  over  46'  below  top  of  exterior  bank.  "Where  the  rock  is  more 
than  46',  two  rows  of  sheet-piling  are  to  be  driven  to  the  rock,  16'  apart,  and  the  material 
between  them  excavated,  so  as  to  remove  all  soil,  muck,  or  vegetable  matter.  Sheet-piling 
will  be  formed  of  spruce  or  pine  plank,  6"  thick,  tongued  and  grooved :  the  tongue  and  groove 
to  be  11"  x  1".  The  earth  within  the  working-lines  of  interior  slopes  will  be  excavated 
to  the  depth  of  40'  below  top  of  exterior  bank,  rock  36'.  The  puddle-ditch  will  be 
formed  of  clay,  gravel,  sand,  or  earth,  or  such  admixture  of  these  materials,  or  any  of 
them,  as  the  engineer  may  direct,  to  be  laid  in  layers  of  not  more  than  6",  well  mixed 
with  water,  and  worked  with  spades  by  cutting  through  vertically,  in  two  courses  at  right 
angles  with  each  other  :  the  courses  to  be  1"  apart,  and  each  spading  to  extend  2"  into 
the  lower  course  or  bed.  Whenever  the  work  is  suspended,  the  puddle  must  be  covered 

*  This  wall  was  formed  of  concrete. 


442  ENGINEERING   DEAWING. 

\vith  boards  or  earth  to  prevent  cracking,  and,  whenever  cracks  do  occur  in  the  puddle, 
those  parts  must  be  removed  and  reworked.  The  puddle  will  extend  to  all  the  masonry 
and  pipes,  and  along  and  around  it  and  them  as  the  engineer  may  direct. 

"  The  embankments  will  be  formed  in  layers  of  not  more  than  6",  well  packed  by 
carting  and  rolling,  and,  in  such  places  as  the  rollers  cannot  be  effectually  used,  by  ram- 
ming. The  embankments  will  be.  worked  to  their  full  width  as  they  rise  in  height  and 
not  more  than  2'  in  advance  of  the  puddle.  The  interior  slopes  of  all  the  banks  will  be 
covered  with  8"  thickness  of  stone,  broken  to  pass  through  a  2"  ring.  On  this  will  be 
laid  the  paving,  18"  in  thickness,  of  a  single  course  of  stones  set  on  edge  at  right  angles 
with  the  slope,  laid  dry,  and  well  wedged  with  pinners." 

Distribution. — Fig.  26  are  plans  of  tlie  various  cast-iron  pipes  used  in  the 
city  of  Brooklyn.  Figs.  27,  28,  29,  30,  31,  and  32,  are  sections  of  the  spigot 
and  faucet  ends  of  some  of  the  same  pipes.  Of  these  pipes  there  were  two 
classes,  A  and  B.  The  A  pipes  were  designed  for  positions  subject  to  an  ex- 
treme head  of  120',  the  B  pipes  for  positions  below  this  level,  subject  to  a  head 
of  from  120  to  170  ft.  It  will  be  observed  (fig.  26)  that  the  4",  6",  and  8" 
pipes  have  belts  cast  on  them.  This  was  to  give  thickness  of  metal  to 
hold  the  tapped  in  branch.  The  thicknesses  adopted  were  greater  than 
given  by  the  English  or  French  rules,  but  consistent  with  the  practice  in 
the  United  States,  and  are  not  now,  after  trial,  considered  too  heavy. 

The  formulae  given  by  Mr.  Seville  are  t  =  .0024  (n  +  10)  d  +  .33  for 
pipes  cast  horizontally,  t  =  .0016  (n  +  10)  d  +  .32  for  pipes  cast  vertically. 

ByM.  Dupuis  of  the  Paris  Water- works,  t  =  .0016  n  d  +  .013  d  +  .32,  in 
which  t  is  the  thickness  in  inches,  n  the  number  of  atmospheric  pressures,  ta- 
ken at  33'  each,  to  which  the  pipe  is  to  be  subjected,  and  d  the  diam.  in  inches. 

For  the  discharge  of  water  through  pipes,  the  formulae  given  by  dif- 
ferent hydraulic  engineers  are  varied.  Experiments  on  a  large  scale  were 
made  by  the  Brooklyn  engineers  on  the  flow  through  some  of  the  Croton 
and  Jersey  City  pipes,  and  a  formula  was  deduced,  which  in  its  simplest 

form  is  expressed —  V  =  40  (^j^j  d ;    and  is  well  adapted  for  ordinary 

purposes,  where  pipe  has  been  laid  some  time,  and  has  been  affected  some- 
what by  corrosion. 

Blackwell's  formula  is  very  nearly  Y  =  48  (^y- \  d. 

V  =  velocity  in  ft.  per  second,  and  all  other  dimensions  being  expressed 
in  ft. ;  thus,  if  the  diameter  of  the  pipe  be  10"  or  .83  ft.,  the  head  100', 

and  the  length  1,000,  then  ^  -  1TW^  Or  l/'°83  =  >289' Y  = 
40  x  .289  =  11.56  ft.  per  second,  and  the  discharge  would  be  the  area  of  a 
10"  pipe,  or  .545  square  ft.  x  11.56  =  6.3  cubic  ft.  per  second. 


ENGINEERING   DRAWING. 


443 


Fig.  26. 


444 


ENGINEERING   DKAWING. 


Fig.  33  is  a  half-plan  and  half-section  of  a  12"  x  8"  4-way  branch,  and 
fig.  34  of  a  36"  sleeve. 


Fig.  33.  Fig.  34. 

In  Plates  XXXVI.,  SXXVIL,  and  XXXVIII.,  are*  given  the  details 
of  one  of  the  large  48"  stop-cocks. 

From  the  specifications  of  "  Cast-iron  Distribution-pipes  and  Pipe- 
mains,  with  their  Branches,"  etc.,  Brooklyn,  L.  I. : 

"  All  pipes  of  20"  diameter  and  upward  to  be  formed  so  as  to  give  a  lead  joint  of  not 
Jess  than  £"  in  thickness  all  round,  and  not  more  than  TV;  those  of  12"  diameter  and 
under,  not  exceeding  f",  and  not  less  than  Ty.  The  straight  pipes  of  12"  diameter  and 
upward  shall  be  cast  in  dry  sand  moulds,  vertically.  The  smaller  pipes  may  be  cast  at 
an  angle  with  the  horizon  of  not  less  than  12°.  The  pipes  shall  be  free  from  scoria,  sand- 
holes,  air-bubble;!,  cold-short  cracks,  and  other  defects  or  imperfections;  they  shall  be 
truly  cylindrical  in  the  bore,  straight  in  the  axes  of  the  straight  pipes,  and  true  to  the  re- 
quired curvature  or  form  in  the  axes  of  the  other  pipes;  they  shall  be  internally  of  the 
full  specified  diameters,  and  have  their  inner  and  outer  surfaces  concentric.  No  plugging 
or  filling  will  be  allowed.  They  shall  be  perfectly  fettled  and  cleaned ;  no  lumps  or  rough 
places  shall  be  left  in  the  barrels  or  sockets.  No  pipes  will  be  received  which  are  defective 
in  joint-room.  The  spigot-ends  of  all  the  branches  to  have  lugs  or  horns  cast  on  each. 
Every  pipe-branch  and  casting  shall  pass  a  careful  hammer-inspection,  and  shall  be  sub- 
ject thereafter  to  a  proof  by  water-pressure  of  300  Ibs.  to  the  square  inch  for  all  pipes 
30"  in  diameter  and  under,  and  250  Ibs.  per  square  inch  for  all  pipe-mains  exceeding  30" 
diameter.  Each  pipe,  while  tinder  the  required  pressure,  shall  be  rapped  with  a  linrul- 
hammer  from  end  to  end,  to  discover  whether  any  defects  have  been  overlooked.  The 
pipes  shall  be  carefully  coated  inside  and  outside  with  coal-pitch  and  oil,  according  to  Dr. 
R.  A.  Smith's  process,  as  follows : 

"  Every  pipe  must  be  thoroughly  dressed  and  made  clean  from  sand  and  free  from  rust. 
If  the  pipe  cannot  be  dipped  presently  after  being  cleansed,  the  surface  must  be  oiled 
with  linseed-oil,  to  preserve  it  until  it  is  ready  to  be  dipped  ;  no  pipe  to  be  dipped  after 


ENGINEERING   DRAWING. 


445 


rust  has  set  in.  The  coal-tar  pitch  is  made  from  coal-tar,  distilled  until  the  naphtha  is  en- 
tirely removed  and  the  material  deodorized.  The  mixture  of  5  or  6$  of  linseed-oil  is  rec- 
ommended by  Dr.  Smith.  Pitch  which  becomes  hard  and  brittle  when  cold  will  not 
answer.  The  pitch  must  be  heated  to  300°  Fahr.,  and  maintained  at  this  temperature 
during  the  time  of  dipping.  Every  pipe  to  attain  this  temperature  before  being  removed 
from  the  vessel  of  hot  pitch.  It  may  then 'be  slowly  removed  and  laid  upon  skids  to 
drip." 

Weights  of  9  and  12  ft.  pipe. 


9  ft.               12  ft. 

1  ft.  laid. 

"Din 

| 

T   r\  '   *  *  t 

A 

B    |     A 

B 

A 

B 

Ibs. 

H>8. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

4' 

200 

38.077 

41.538 

6' 

330 

360 

. 

49.615 

57.692 

8.5 

8' 

430 

500 

570« 

660 

76.286 

92.556 

10. 

12' 

890 

1,080 

16. 

20' 

2,100 

2,500 

30.3 

30' 

3,960 

4,890 

53.5 

36' 

4,750 

69.5 

48' 

8,300 

For  the  removal  of  waste  water  and  raini'all  from  houses,  sewers  are 
very  convenient  in  towns  and  cities,  even  before  the  construction  of  water- 
works; but  after  the  introduction  of  a  liberal  arid  regular  supply  of  water, 
sewers  are  indispensable  in  removing  this  water  after  it  has  been  used. 
The  ruling  principle  in  the  establishment  of  sewerage-works  is,  that  each 
day's  sewage  of  each  street  and  of  each  dwelling  should  be  removed  from 
the  limits  of  city  and  town,  as  far  as  practicable,  on  the  day  of  its  produc- 
tion, that  it  should  pass  off  before  decomposition  begins,  that  it  should  not 
be  allowed  to  settle  and  fester  in  the  sewers,  producing  those  noxious  gases 
so  prejudicial  to  health.  To  attain  this  end,  the  refuse  fluids  must  be  suf- 
ficient in  quantity  to  float  and  carry  off  the  heavier  matters  of  sewage. 

If  the  rate  of  inclination  of  a  sewer  be  not  less  than  1  in  440,  the  ex- 
perience of  Brooklyn,  and  other  cities  equally  well  supplied  with  water, 
shows  that  the  fluid  domestic  sewage  is  sufficient  to  carry  off  all  the 
heavier  matters,  and  keep  the  sewers  free  and  clean,  provided  the  form  is 
such  as  to  concentrate  as  much  as  possible  the  sewage  waters.  Less  in- 
clination than  1  in  440  will  require  some  means  of  flushing.  The  Brook- 
lyn system  of  sewers,  adopted  on  the  report  and  plans  of  Mr.  J.  "W.  Adams, 
has  been  as  successful  as  any  that  I  know  of,  and  is  entirely  different  from 
the  former  practice  in  this  country :  the  principle  of  construction  has  been, 
to  make  the  sewers  as  small  as  the  service  required  of  them  will  admit,  to 
maintain  as  much  velocity  of  flow  as  possible,  so  that  nothing  may  be  de- 
posited, without  any  provision  for  a  man  entering  and  passing  through  the 


416 


ENGINEERING   DRAWING. 


sewer,  which  has  been  found  by  experience  unnecessary.  The  conditions 
which  govern  the  size  of  the  sewer  are  the  quantity  of  sewage  to  be  dis- 
charged, the  quantity  of  rainfall,  and  the  inclination  of  the  sewer ;  but 
household  sewage  is  small  in  comparison  with  the  extremes  of  rainfall. 
The  rule  adopted  by  Mr.  Adams  has  been  that  given  by  Mr.  J.  ~\V.  Bazal- 
gette,  engineer  of  the  London  drain  age- works,  viz. : 
3  log.  A -flog,  y  +  6.8 


10 


=  log.  D  ; 


or,  more  mathematically  expressed, 

D  =  ( A3  x  NX  6309574)  £: 

in  which  A  is  the  area,  in  acres,  to  be  drained ;  !N",  the  distance,  in  feet,  in 
which  the  sewer-pipe  falls  1  ft. ;  and  D  the  diameter  in  inches. 

The  value  of  sewers  depends  on  the  correctness  of  their  lines,  uniformity 
of  descent,  and  smoothness  of  interior  surface.  The  pipes  used  at  Brook- 
lyn have  generally  been  strong  glazed  earthenware  pipes  of  12",  15",  and 
18"  diameter.  Many  cement  pipes  have  also  been  used,  and,  in  such  situa- 
ations  as  required  great  capacity,  brick  sewers  were  used,  the  leading 
forms  of  which  are  egg-shaped,  as  in  fig.  35,  of  which  the  dimensions 

are  as  follows,  the  longest  diameter  D  and 
the  longest  radius  R'  being  alike  in  each 
size,  and  E"  $  R,  and  D  3  times  R: 


Area. 

R 

! 
D 

60"  diameter  circular 

24:808      1     74.425 

48"        "              " 

19.8465 

59.54 

36"        "              " 

14.886 

44.  Goo 

24"        "              " 

9.923 

29.77 

Thickness   of  brickwork,  8";    boards 
shown  at  bottom  only  used  in  cases  of 
soft  earth  for  convenience  of  construction. 
Rule  to  determine  area  of  egg-shaped  sewer  of  above  section : 

=  4.5942  R2,  or  nearly  multiply  R2  x  4.6. 

In  some  locations  the  depth  did  not  admit  of  the  egg-shaped  sectio'n. 
A  circular  form  of  6  ft.  diameter  was  adopted  for  the  Union  Avenue 
sewer,  and  one  of  a  section  similar  to  the  main  conduit  of  the  water-works 
10  ft.  in  width  and  9  ft.  high,  in  the  clear,  for  Kent  Avenue. 

Man-holes  are  built  along  the  line  of  sewers,  at  a  distance  of  from  100 
to  150  ft.  apart,  to  give  access  to  the  sewers  for  purposes  of  inspection  and 
removal  of  deposit. 

Figs.  36  and  37  are  section  and  plan  of  the  man-hole  at  present  used  by 


ENGINEERING-   DEAWIXG. 


the  Croton  Sewer  Department.  It  consists  of  a  funnel-shaped  brick  well, 
oval  at  the  bottom,  4'  x  3',  circular  at  top,  2'  diameter,  curbed  with  cast- 
iron  frame  and  covered  by  cast-iron  plate.  Side-walls,  8"  thick,  through 
which  the  pipe-sewers  pass  at  the  bottom  of  the  well.  Across  the  open 
space,  the  sewer  is  formed  in  brick,  whose  bottom  section  corresponds  to  that 


Fig.  30. 


Fig.  37.— Scale:  J"  =  1  ft. 

of  pipe,  side-walls  carried  up  perpendicular  to  top  of  sewer ;  the  flat  spaces 
at  the  sides  of  sewer  are  nagged.  In  the  figure  the  main  sewer  is  12" 
pipe,  with  a  12"  branch  entering  at  an  acute  angle,  as  all  branches  and 
connections  with  a  sewer  should.  The  short  lines  on  the  left  vertical  wall 
represent  sections  of  f|  staples,  built  in  to  serve  for  a  ladder.  In  the  Brook- 
lyn sewer  the  covers  of  these  man-holes  were  at  first  perforated  for  the  venti- 
lation of  the  sewer,  but  this  has  proved  to  be  unnecessary,  and  they  are 
now  made  close.  If  a  sewer  performs  the  function  for  which  it  is  in- 
tended, of  removing  all  waste  before  it  becomes  offensive,  ventilation  is 
not  necessary. 


ENGINEERING   DRAWING. 


Wherever  necessary,  from  the  slope  and  confirmation  of  the  ground, 
to  remove  the  surface  or  rain  water  directly  from  the  street-gutters  into 
the  sewers,  catch-basins  are  placed  generally  at  the  corners  of  streets. 

Figs.  38  and  39  are  section  and  plan  of  the  Croton  sewer  catch-basins,  on 
a  scale  of  \"  —  1  ft.  The  intention  of  the  catch-basin  is  to  receive  the 


street  washings,  retain  the  heaviest  portion  in  the  basin,  and  let  the  liquid 
escape  into  the  sewer.  The  basin  in  the  figure  is  rectangular  in  plan,  with 
a  semicircular  end,  3'  8"  in  width  by  5'  V  long ;  bottom  of  flag,  and  side- 
walls  of  brick  12"  thick.  It  will  be  observed  that  a  piece  of  flag  is  built 
into  the  side-walls  from  the  top,  extending  about  half  way  to  the  bottom ; 
this  divides  the  upper  part  of  the  basin  into  two  parts  ;  the  sewer  enters 
the  basin  3  ft.  above  bottom  flag ;  the  dividing  flag  comes  to  within  2'  6" ; 
before  any  water  can  flow  out  through  the  sewer-pipe  this  flag  must  be 
submerged  6" ;  a  trap  is  thus  formed,  which  cuts  off  any  smell  from 


ENGINEERING   DRAWING.  449 

the  sewer  escaping  into  the  street.  The  water  is  received  into  the 
basin  through  the  channel  C,  which  is  curbed  with  granite,  and  protected 
4)y  a  grating.  The  coping  (b)  is  of  granite,  and  forms  a  portion  of  the 
sidewalk;  through  this  there  is  a  man-hole  cut,  16"  diameter,  for  access 
to  the  basin,  and  for  removal  of  the  deposit ;  it  is  covered  by  a  strong  cast- 
iron  plate. 

Gas  Supply. — !N"ext  in  importance  to  the  necessities  of  a  city  or  town 
for  water  supply  and  sewerage,  is  the  luxury  of  gas  supply.  The  gas-works 
should  always  be  placed  remote  from  the  thickly-populated  part  of  a  city, 
for  under  the  best  regulations  some  gas  will  escape  in  the  manufacture, 
offensive  and  deleterious.  They  should  be  placed  at  the  lowest  level,  for, 
gas  being  light,  readily  rises,  and  the  portions  of  the  city  below  the  works 
are  supplied  at  less  pressure  than  those  above.  Gas-mains,  like  those  for 
water,  are  of  cast  iron,  and  put  together  in  the  same  way ;  but,  as  they 
have  to  resist  no  pressure  beyond  that  of  the  earth  in  which  they  are 
buried,  they  are  never  made  of  as  great  thickness  as  those  of  water-pipes. 

Weight  of  Gas-pipes  per  running  foot. 


The  quantity  of  gas  that  can  be  delivered  through  a  main  depends  on 
the  size  and  length  of  the  pipe,  and  on  the  specific  gravity  of  the  gas  and 
the  pressure  with  which  it  is  driven  through  the  mains.  This  last  is  usu- 
ally estimated  in  inches  of  water.  The  specific  gravity  ranges  from  .4  to 
.5,  atmospheric  air  being  1.  The  pressure  is  reduced  by  friction  and  leak- 
age of  pipes,  and  increases  or  diminishes  about  T^-g-  of  an  inch  for  every 
foot  rise  or  fall  in  the  level  of  the  pipe. 

The  usual  house-burner  consumes  about  4  cubic  ft.  per  hour.  Large  or 
Argand  burners,  from  6  to  10  cubic  ft. 

In  the  formula,  D  =  0  \/  r±-5L  for  determining  the  size  of  the  small 

wrought  iron  for  the  distribution  throughout  dwellings  and  street-mains, 
the  value  of  C  varies  from  .073  to  .063.  Without  illustrating  by  an  exam- 
ple, it  will  be  sufficient  for  the  purpose  to  give  a  calculated  discharge  for 
different-sized  pipes  of  one  length  and  at  one  pressure,  merely  stating  that 
the  discharge  will  vary  directly  as  the  square  root  of  the  pressure,  and 
inversely  as  the  square  root  of  the  length,  i.  e.,  under  a  pressure  of  two 
inches  the  same  pipe  will  deliver  1.41  (/2)  times  as  much  gas  as  under  a 


12  Ibs. 

10"  

50  Ibs. 

16     " 

12"  

62     " 

27     " 

16"  

103     " 

...40     " 

20"... 

..150     " 

450  ENGINEERING   DRAWING. 

pressure  of  one  inch,  and  under  the  same  pressure,  but  with  4  times  the 
length,  the  same  size  of  pipe  will  deliver  one-half  I  —  I  as  much  gas. 

Gas  Service-Pipes. — Under  a  pressure  of  -3^-  of  an  inch,  and  with  9 
length  of  pipe  100  ft., 

a  pipe  of  diameter    f  "       %"        f  "       I"        1J"        l^" 
will  deliver  8.6     17.5     48.3     99.     173.3     272.6  cubic  ft.  per  hour. 

Gas-Mains. — Under  a  pressure  of  one  inch,  and  with  a  length  of  pipe 
of  10,000  ft., 

a  pipe  of  diameter  of          3"  4"  6"  9"  12" 

will  deliver  per  hour  401.8  826.2  2276.  6274.  12879  cubic  ft. 
Itoads. — Under  this  general  term  are  included  all  routes  of  land-travel ; 
but  the  term  "  streets  "  is  applied  mostly  to  city,  town,  and  suburban  roads, 
while  "roads"  and  "highways"  are  applied  to  those  of  the  country.  By 
an  "  avenue  "  is  generally  understood  a  wide  street ;  although  in  this  city  all 
the  streets  designated  by  numbers  or  letters,  running  north  and  south,  are 
called  avenues,  and  those  at  right  angles,  streets.  The  term  boulevard  is 
applied  to  very  wide  avenues  in  which  there  are  rows  of  trees.  The 
term  street  in  this  city  is  legally  applied  to  that  portion  which  cannot  be 
built  on — that  is,  the  street-lines,  as  laid  out,  are  the  established  bounds  on 
which  buildings  may  be  erected.  The  street,  therefore,  technically  in- 
cludes the  street  or  travelled  way 
for  carriages,  and  the  sidewalks  and 
area.  Our  numbered  streets  are 
usually  60  ft.  wide— that  is,  60  ft.  be- 

rlKl   tween  brick-lines— of  this  30  ft.  are 
.-  .  .  .,,..,.. 

devoted  to  carriages,  and  15  ft.  on 
each  side  to  footwalks  and  areas ;  the 

avenues  and  wide  streets  are  each  100  ft.  wide,  with  60  ft.  for  carriages  and 
20  ft.  on  each  side.  The  space  occupied  by  areas,  or  from  street-line  to 
line  of  fence,  is  usually  5  ft. ;  this  may  be  enclosed  by  iron  fence,  but  can- 
not be  included  within  the  building  above  the  level  of  sidewalk.  The 
stoop-line  extends  into  the  sidewalk  beyond  the  area-line  some  V  to  18", 
fixing  the  limit  for  the  first  step  and  newel  to  a  high  stoop  or  platform. 
The  new  boulevard  which  is  to  occupy  the  present  line  of  Broadway  and 
the  Bloomingdale  Koad  is  150  ft.  wide,  of  which  100  ft.  are  to  be  carriage- 
way, and  25  ft.  on  each  side  for  sidewalk  and  area,  the  latter  not  to  exceed 
7  ft, ;  one  row  of  trees  to  be  set  within  the  sidewalk,  about  2  ft,  from  the 
curb.  In  Paris,  there  is  no  area;  the  sidewalk  comes  up  to  the  house 
or  street-line,  and  there  is  a  space  for  trees  between  sidewalk  and  street- 


ENGINEERING   DRAWING. 


451 


curb.  This  space  is  available  for  pedestrians,  a  part  being  a  gravel,  as- 
phalt, or  flagged  walk.  The  following  are  the  dimensions  according  to  the 
law  of  June  5,  1856 : — 


Entire  width                 Width 
of  boulevard  and                 of 

Width 
of 

Width  for  trees  !        of 

Distance  of  row  from 

avenues. 

carriage-way.    :  sidewalk. 

j     trees. 

Street-line. 

St.-curb. 

Metres. 

Metres. 

Metres. 

Metres. 

Metres. 

Metres. 

26  to  28 

12 

1 

5.5  to  6.5 

1.5 

30    "    34 

14 

1        j    6.5    "   8.5 

1  5 

36    »    38 

12  to  13 

3.5 

8.  to  8.5             2- 

5.      "   5.5 

1.5 

40 

14 

3.5 

9.5                  2 

6.5 

1.5 

1  metre  =  3.281  ft. 


eet- 


The  footwalks  in  this  city  and  vicinity  are  generally  formed  of  flags, 
or  what  is  here  termed  blue-stone,  laid  on  a  bed  of  sand  or  cement-mortar. 
The  flags  are  from  2"  to  4"  thick.  In  the  more  important  streets  the  up- 
per surface  is  axed,  the  quality  of  the  stone  selected,  and  the  dimensions 
are  sometimes  as  much  as  10'  X  14'.  Brick  are  often  used  in  towns,  or 
places  where  good  flag  cannot  be  readily  obtained,  usually  laid  flatways  on 
a  sand-bed.  Granite  is  very  often  employed  in  business  streets,  in  lengths 
the  full  width  of  the  sidewalk,  and  about  1'  in  thickness,  the  inner  ends 
resting  on  a  cast-iron  girder,  and  the  outer  on  the  vault  wall,  forming  in 
this  way  a  roof  for  the  vault,  and  the  outer  ends  a  curb  for  the  street. 

Curb.       Sidewalk. 


Carriage-wa 


Fig.  42. 


Curbs  here  are  generally  of  flag,  about  4"  thick  by  20"  deep,  extending 
10*  above  the  gutter-stone ;  but  where  the  street  is  nearly  level,  and  the 


452  ENGINEERING   DRAWIXG. 


gutter-stones  have  to  be  raised  to  give  sufficient  descent  for  the  flow  of  the 
water,  the  curbs,  in  extreme  cases,  are  not  more  than  4"  exposed.  The 
gutter-stones  are  from  12"  to  15"  in  width,  and  from  3"  to  5"  in  thickness, 
laid  close,  and  bedded  in  cement.  The  bridge  or  crossing  stones  are  of 
blue-stone  or  granite,  from  2'  to  15"  wide,  and  not  less  than  5"  thick,  laid 
in  double  rows. 

Carriage-way.  —  Most  of  our  streets  and  avenues  are  paved  —  formerly 
entirely  of  cobble-stone,  and,  if  selected  so  as  to  be  of  a  uniform  size  and 
shape,  and  properly  bedded  in  sand,  and  well  rammed,  they  form  in  many 
places  a  cheap  and  very  fair  roadway  ;  but  the  cubical  block  stone  pave- 
ment of  trap  or  granite,  often  called  the  Belgian,  is  -in  every  way  to  be 
preferred.  Mr.  Kneass,  the  engineer  of  the  city  of  Philadelphia,  recom- 
mends 

';  That  the  blocks  should  not  exceed  3"  in  width,  6"  in  depth,  nor  8"  in  length  ; 
that,  as  to  depth,  they  should  be  uniform  within  £",  and  in  length  be  not  less  than  6". 
For  foundations  the  material  should  be  taken  out'  to  a  depth  of  20"  below  the  proposed 
surface  of  paying,  and  to  be  made  to  accurately  conform  to  shape  of  finished  road. 
After  being  compactly  rolled  with  a  heavy  roller,  it  should  have  a  covering  of  clean 
anthracite  coal-ashes,  placed  upon  it  to  a  depth  of  10",  laid  on  in  two  layers,  each  well 
rolled  ;  the  ashes  to  be  scrupulously  clean,  i.  e.  free  from  any  organic  matter.  Upon 
this  should  be  laid  a  bed  of  clean  gravel,  4"  in  depth,  and  rolled  ;  upon  which  again 
should  be  a  layer  of  sand,  clean  and  sharp,  or  fine-screened  gravel,  in  which  to  set  and 
bed  the  stone-blocks.  Each  layer  of  ashes  and  gravel  should  in  surface  conform  to  the 
outline  intended  for  the  surface  of  the  stone.  The  stone  should  be  carefully  assorted, 
so  that,  when  laid  across  the  street,  the  joint-lines  may  be  straight  ;  and  each  stone 
should  be  set  on  its  led  fair  and  square,  so  that  no  edge  shall  extend  above  the  general 
level  of  the  surface,  and  the  surface  of  each  stone  shall  be  an  extension  of  that  lying 
next  to  it.  The  joints  I  would  not  make  smaller  than  £",  to  be  filled  with  sand  and 
grouted  with  liquid  lime.  Before  grouting,  the  entire  surface  should  be  rammed  until 
no  impression  can  ~be  made  on  it." 

Broadway,  as  far  as*  Fourteenth  Street,  was  originally  paved  with 
Buss  pavement,  consisting  of  blocks  of  granite,  9"  to  12"  wide  X  10"  to 
15"  long  X  10"  deep,  in  diagonal  rows,  resting  on  a  concrete  base,  with 
sand  enough  merely  to  bring  the  blocks  to  a  uniform  surface.  These 
blocks  are  now  being  replaced  by  granite,  in  rows  at  right  angles  to 
street,  from  4"  to  5"  wide,  8"  to  15"  long,  and  9".  deep,  with  a  bed  of  3" 
sand  on  the  old  concrete  base.  "  In  London,  the  usual  practice  is,  to  set 
their  blocks  3"  wide  X  9"  deep,  and  from  6"  to  12"  long,  on  a  bed  of 
gravel,  with  a  base  of  broken  granite  12"  deep. 

Wooden  pavements  have  been  tried  of  various  kinds,  but  the  only  one 
in  any  extensive  use  is  the  JSTicolson  patent,  which  consists  of  pieces  of 


ENGINEERING   DRAWING.  453 

3"  plank,  G"  long,  set  on  a  board  base  supported  by  a  sand-bed.  The 
plank  is  set  on  end  in  lines  perpendicular  to  line  of  street,  with,  a  strip  of 
board  V  wide  between  the  rows,  nailed  to  the  blocks :  the  top  of  strip 
being  some  2"  to  3"  below  top  of  plank.  Boiled  coal-tar  is  used  Jreely 
while  setting  the  blocks,  and  is  poured  into  the  interstices  ;  the  V  joint  is 
filled  with  gravel,  wet  with  tar,  and  well  rammed. 

In  Paris,  of  late  years,  asphalt  has  been  used  to  a  very  great  extent, 
both  for  foot  and  carriage  ways.  The  carriage-ways  are  composed  of  a 
layer  of  asphalt,  from  lijr"  to  2"  thick,  on .  a  bed  of  concrete,  or  on  a  worn 
McAdam  road,  over  which  is  spread  a  thin  coat  of  cement.  The  cement 
having  become  dry,  the  asphaltic  rock,  reduced  to  a  powder,  is  spread  over 
the  surface  to  a  depth  of  about  .40  per  ct.  more  than  the  finished  thickness ; 
it  is  then  rammed  with  rammers  warmed  by  portable  furnaces,  beginning 
gradually,  and  increasing  the  force  of  the  blows  as  the  work  approaches 
completion.  For  a  footway  the  same  concrete  bed  is  used,  and  the  layer 
of  asphalt  is  about  f".  "Walks  and  roads  have  been  constructed  in  this 
country  with  an  artificial  asphalt,  prepared  from  coal-tar  mixed  with 
gravel.  None  have  yet  stood  the  test  of  time  sufficiently. 

Roads  and  Highways. — McAdam  first,  in  England,  made  the  con- 
struction of  broken-stone  roads  a  science,  and  has  given  his  name,  in  this 
country,  to  all  this  class  of  roads.  lie  says  that  the  whole  science  of  arti- 
ficial road-making  consists  in  making  a  dry  solid  path  on  the  natural  soil, 
and  then  keeping  it  dry  by  a  durable  water-proof  covering.  The  road- 
bed, having  been  thoroughly  drained,  must  be  properly  shaped,  and  sloped 
each  way  from  the  centre,  to  discharge  any  water  that  may  penetrate  to  it. 
Upon  this  bed  a  coating  of  3"  of  clean  broken  stone,  free  from  earth,  is  to 
be  spread  on  a  dry  day.  This  is  then  to  be  rolled,  or  worked  by  a  travel 
till  it  becomes  almost  consolidated;  a  second  3"-layer  is  then  added,  wet 
down  so  as  to  unite  more  readily  with  the  first ;  this  is  then  rolled,  or 
worked,  and  a  third  and  fourth  layer,  if  necessary,  added.  Mc^\  dam's 
standard  for  stone  was  6  oz.  for  the  maximum  weight,  corresponding  to  a 
cube  of  1£",  or  such  as  would  pass  in  any  direction  through  a  2%"  ring. 
The  Telford  road  is  of  broken  stone,  supported  on  a  bottom  course  or  layer 
of  stone  set  by  hand  in  the  form  of  a  close,  firm  pavement. 

At  our  Central  Park,  after  trials  of  the  McAdam  and  Telford  roads, 
the  grayel-road  (of  which  fig.  43  is  a  cross-section  of  one-half)  was  adopted, 
as  being,  according  to  the  statement  of  their  engineer,  Mr.  Grant,  "  the 
easiest  and  most  agreeable  kind  of  road  for  both  carriages  and  horses  ;  that 
it  is  the  cheapest  at  first  cost,  and  can  be  kept  in  repair  at  an  equal  if  not 
less  cost  than  any  other  equally  satisfactory  road."  This  road  consists  of 


454 


ENGINEERING   DRAWING. 


a  layer  of  rubble-stones,  about  7"  thick,  on  a  well-rolled  or  packed  bed, 
with  a  covering  of  5"  of  gravel.  C  are  the  catch-basins  for  the  reception 
of  water  and  deposit  of  silt  from  the  gutters ;  S  is  the  main  sewer  or 
drain,  mnd  s  a  sewer-pipe  leading  to  catch-basin  on  opposite  side  of  the 


Fig.  43. 

road.  In  wider  roads  each  side  has  its  own  main  drain,  and  there  is  no 
cross-pipe  s.  The  road-bed  was  drained  by  drain-tiles  of  from  1^-"  to  4"- 
bore,  at  a  depth  of  3'  to  3|-'  below  the  surface.  The  maximum  grade  of 
the  Park  roads  is  1  in  20.  The  grades  of  the  streets  of  Paris  vary  from  1 
in  20  to  1  in -200.  The  best  grade  is  from  1  in  50  to  1  in  100 ;  this  gives 
ample  descent  for  the  flow  of  water  in  the  gutters.  Many  of  our  street- 
gutters  have  a  pitch  not  exceeding  1  ft.  in  the  width  of  a  block,  or  200  ft. 

The  grade  of  a  road  is  described  as  1  in  so  many — so  many  feet  to  the 
mile,  or  such  an  angle  with  the  horizon. 


Inclination. 
1  in  10 

Feet  per  mile. 
528 

Angle. 
5°  43' 

I 

iclination. 
in    30 

Feet  per  mile. 
176 

Angle. 
1°  55' 

1    "   11 

462 

5° 

"     40 

132 

1°  26' 

1    "   14 

369 

4° 

"     50 

106 

1°    9' 

1    "   20 

264 

•    2°  52' 

"     57 

92 

1° 

1    "   29 

184 

2° 

"  100 

53 

35" 

The  best  transverse  profile  for  a  road  on  nearly  level  ground  is  that 
formed  by  two  inclined  plains,  meeting  in  the  centre,  and  having  the  angle 
rounded.  The  degree  of  inclination  depends  somewhat  on  the  surface  of 
the  road.  A  medium  for  broken-stone  roads  is  about  \"  in  1',  or  1  in  24 ; 
but  Telford,  on  the  Ilolyhead  Road,  adopted  1  in  30 ;  and  McAdam,  1  in 
3G,  and  even  1  in  60.  For  paved  streets  in  Paris,  a  crown  of  •£$•  of  the 
width  is  adopted,  and  for  McAdamized,  -3-^.  The  inclination  of  sidewalks 
should  not  exceed  $"  in  1  ft.,  and,  when  composed  of  granite,  the  surface 
should  be  roughened. 

The  necessity  of  a  well-drained  road-bed  is  as  important  beneath  a  rail 
as  on  a  highway.  The  cuts  should  be  excavated  to  a  depth  of  at  least  2 


ENGINEERING   DRAWING.  455 

ft.  below  grade,  with  ditches  at  the  sides  still  deeper,  for  the  discharge  of 
water.  The  embankments  should  not  be  brought  within  2  ft.  of  grade : 
this  depth  to  be  left  in  cut  and  on  embankment  for  the  reception  of  ballast. 
The  best  ballast  is  McAdam  stone,  on  which  the  cross-ties  are  to  be 
bedded,  and  finer-broken  stone  packed  between  them.  Good  coarse  gravel 
makes  very  good  ballast ;  but  sand,  although  affording  filtration  for  the 
water,  is  easily  disturbed  by  the  passage  of  the  trains,  a  great  annoyance 
to  travellers,  and  injury  to  the  rolling  stock  by  getting  into  boxes  and 
bearings.  The  average  length  of  sleepers  on  the  4.8£  gauge  railways  is 
about  8  ft. ;  bearing  surface,  1" ;  distance  between  centres,  2'  4".  English 
railways,  9  ft.  long,  10"  wide,  5"  thick,  3'  centres,  and  on  lines  of  very 
heavy  traffic,  2'  6"  centres.  Average  width  of  New- York  railways,  of  same 
gauge  as  above,  for  single  lines,  in  cuts  IS',  banks  13' ;  for  double  lines, 
cuts  31',  banks  26£'.  The  width  between  two  lines  of  track  was  formerly 
6  ft. ;  but,  with  the  increase  in  overhang  of  cars,  it  is  often  now  made  7  ft. 

Resistance  of  WJieel- Carriages  on  Roads. 


Soft,  gravelly  ground \  =  .143 

Gravel  road ^  =  .067 

Broken  stone  on  firm  foundation .  J-  =  .029 


Stone  pavement -fa  =  .015 

On  level  railway  track,  and  be- 
low speed  of  10  miles  per  hour  -^^  =  .003 


Sledge  on  hardened  snow,  about  J$. 

The  tractive  force  which  a  horse  can  exert  steadily  and  continuously,  at 
a  walk,  is  estimated  at  about  120  Ibs. 

There  are  many  circumstances  in  the  condition  of  roads  and  of  car- 
riages which  may  reduce  these  resistances  a  little,  but  more  probably  in 
practice  very  much  increase  them.  Below  a  speed  of  10  miles  per  hour, 
the  resistance  of  railway  trains  represents  only  the  effect  of  friction ;  above 
it,  the  concussion  and  resistance  of  the  air  must  come  into  the  calculation. 

Bridges. — Fig.  44  is  an  elevation  of  a  pile  pier  for  a  bridge.  Tenons 
are  cut  on  the  top  of  the  piles,  and  a  cap  (a)  mortised  on.  The  two  outer 
piles  are  driven  in  an  inclined  position,  and  the  heads  bolted  to  the  piles 
adjacent.  The  piles  are  made  into  a  strong  frame  laterally  by  the 
planks  b  and  c,  and  plank  braces  d  d  on  each  side  of  the  piles,  bolted 
through.  The  string-pieces  of  the  bridge — of  dimensions  adapted  to  the 
traffic,  and  the  distance  between  the  sets  of  piles — rest  011  the  cap.  Longi- 
tudinal braces  are  often  used,  their  lower  ends  resting  on  the  plank  b 
— which  should  be,  then,  notched  on  to  the  piles — and  their  upper  ends 
coming  together,  or  with  a  straining-piece  between,  beneath  the  string- 
pieces,  acting  not  only  as  supports  to  the  load,  but  also  as  braces  to  pre- 
vent a  movement  forward  of  the  frames ;  as  the  tendency  of  a  moving  train 
is,  to  push  the  structure  on  which  it  is  supported  forward,  in  railway  bridges 


456 


ENGESEEBIXG   DRAWING. 


especially,  great  care  is  taken  to  brace  the  structure  in  every  way — vertically 
and  horizontally,  laterally  and  longitudinally.  If  the  plank  c  be  a  timber- 
sill,  and  the  piles  beneath  be  replaced  by  a  niasonry-pier,  the  structure 
will  represent  a  common  form  of  trestle. 

rr      rr        n         no 


Fig.  45  is  a  plan  of  one  of  the  stone  piers  of  the  railway  bridge  across 
the  Susquehanna,  at  Havre  de  Grace.  To  lessen  as  much  as  possible  the 
obstruction  to  the  flow  of  the  stream,  and  increase  of  the  velocity  of  current 


in  the  spaces,  which  might  wash  out  the  foundations  of  the  piers  them- 
selves, it  is  usual  to  make  both  extremities  of  the  piers  pointed.  Some- 
times the  points  are  right  angles ;  sometimes,  angles  of  60°  ;  often,  a  semi- 
circle, the  width  of  the  pier  being  the  diameter ;  occasionally,  pointed 
arches,  of  which  the  radii  are  the  width  of  the  pier,  the  centres  being  alter- 
nately in  one  side,  and^heir  arcs  tangent  to  the  opposite  side.  It  will  be 
observed  (fig.  45)  that  none  of  the  stones  break  joint  at  the  angle — this  is 
important  in  opposing  resistance  to  drift-wood  and  ice.  "When,  from  its 
position,  the  pier  is  liable  to  the  blows  of  strong  field-ice,  the  upper  point 
of  the  pier  should  be  formed  with  a  long  incline,  so  that  the  ice  may  be 
raised  up  out  of  the  water,  and  break  by  its  own  weight.  It  is  not  unusual, 
in  very  exposed,  places,  to  make  distinct  ice-breakers  above  each  pier — usu- 
ally of  strong  crib-work,  with  a  plank-slope  like  a  dam,  of  45°,  and  wtih 


ENGINEERING    DRAWING. 


457 


a  width  somewhat  more  than  that  of  the  pier — a  cheap  structure,  that  may 
preserve  a  costly  bridge. 

Bridges  may  be  divided  into  three  general  classes : 

1st.  Arch  bridges — whether  of  stone,  brick,  or  metal — wherein  the 
parts  of  the  arch  exert  a  direct  thrust  upon  the  abutments,  resisted  by  the 
inherent  weight  of  the  latter,  or  its  absolute  fixed  mass,  as  in  the  case  of 
natural  rock  abutments. 

2d.  Girders  or  truss  bridges,  which  exert  no  thrust  on  the  abutments, 
but  rest  thereon  by  their  weight  simply,  the  combined  action  of  thrust  and 
tension  by  which  the  bridge  is  sustained  in  place  being  wholly  within  the 
framing  of  the  bridge. 

3d.  Suspension  bridges,  wherein  the  weight  rests  upon  chains  and 
cables,  the  latter  being  secured  to  the  natural  abutments,  or  the  tension 
strain  resisted  by  masses  of  masonry,  to  which  the  chains  are  anchored. 

The  calculations  required  for  determining  the  stress  and  strains,  in  the 
first  case,  are  among  the  most  delicate  researches  of  applied  mathematics. 
The  theory  of  Coulomb  best  fulfils  the  varying  conditions  which  are  found 
to  exist  in  practice,  but  they  are  much  too  oomplicated  for  insertion  in  the 
present  work ;  and  it  is  considered  of  more  value  to  give  dimensions  of 
existing  bridges,  with  some  empirical  rules  determined  from  them. 


Location. 

Material.  !  Form  of  Arch. 

Span. 

Rise. 

Depth 
at 
crown. 

Depth 
at 
spring. 

High  Bridge,  Harlem  River,  N.  Y.. 
Orleans  and  Tours  R  R 

Stone 

U 

(( 

Brick 

u 

Stone 

Semicircular 

u 

Segmental 
Semicircular 

Segmental 
Elliptical 
Segmental 
Semicircular 

it  ' 

Segmental 
Elliptical 

Segmental 

80 
27.7 
82.5 
24.4 
13.2 
60. 
65. 
79. 
18. 
63. 
30. 
87. 
128. 

124. 

40. 

13.5 
12.2 
6.6 
13.6J 
21. 
13.6 
9. 
31.6 
15. 
16. 
24.3 

6.11 

2.8 
2.7i 
4.6 
1.4 
1.7i 
3.6 
3.9 
3.6 
1.6 
3. 
1.6 

4.H 

5. 

2.8 

Uniform 

4.6 
7.4 
4.6 
Uniform 

2.3 
Uniform 

7.H 

3.7 

Chemin  du  Fer  du  Nord,  sur  1'Oise. 
D'Enghien  R.  R.  du  Nord  
Du  Crochet  R  R 

Stirling  Bridge               

Carlisle       "      
Hutcheson  " 

Manchester  and  Birmingham  R.  R.  . 

u               a                  u                   a 

London  and  Brighton  R.  R  
"          "    Blackwall    "     

Great  Western  R.  R  
Experimental    arch,  designed   and 
built  by  M.  Vaudray,  Paris  

The  arch  last  in  the  list  was  a  very  bold  specimen  of  engineering,  built 
as  an  experiment,  preliminary  to  the  construction  of  a  bridge  over  the 
Seine.  It  was  made  of  cut  stone,  laid  in  Portland  cement,  with  joints  of 
f ",  and  left  to  set  four  months ;  the  arch  was  12'  wide ;  the  centres  rested 
on  posts  in  wrought-iron  boxes  filled  with  sand,  and,  as  the  centring  was 
eased  by  the  running  out  of  the  sand,  the  crown  came  down  -fa* ;  the 


458  ENGINEERING    DRAWING. 


joints  of  one  of  the  skew-backs  opening  y^/'  during  the  first  day,  it  came 
down  -j-L".  It  was  then  loaded  with  a  distributed  weight  of  300  tons  ; 
under  this  load  the  crown  settled  ^"  more.  Since  then  nothing  has  stirred, 
although  it  was  afterward  tested  by  allowing  5  tons  to  fall  vertically  V  6" 
on  the  roadway  over  the  keystone.  This  bridge  will  not  come  within  any 
of  the  rules  laid  down  for  other  constructions.  It  will  be  observed  that 
the  rise  is  about  -fa  the  span,  although  the  usual  practice  for  segmental  and 
elliptical  arches  is  more  than  -jl,  or  within  the  limits  of  J  and  -J-. 

To  determine  the  depth  of  the  keystone,  Rankin.  gives  the  following 
empirical  rule,  which  applies  very  well  to  most  of  the  above  examples  : 
Depth  of  keystone  for  an  arch  of  a  series,  in  feet,  =  4/.17  x  radius  at  crown. 
For  a  single  arch,  =  4/.12  x  radius  at  crown. 

To  find  the  radius  at  crown  of  a  segmental  arch,  add  together  the 
square  of  half  the  span  and  the  square  of  the  rise,  and  divide  their  sum 
by  twice  the  rise  — 


Thus,  the  Blackwall  Railway-bridge  has  a  span  of  87  ft.,  and  a  rise  of  10  — 

43^+16^  _  1892.25  +  256  _  6-  -, 
2x16  ~~32~ 

To  find  the  radius  of  an  elliptical  arch,  we  proceed  on  the  hypothesis 
that  it  is  an  arch  of  5  centres  (fig.  164,  p.  72),  then  the  half  span  is  a  mean 
proportional  between  the  rise  and  the  radius.     Thus,  for  example,  the 
Great  Western  Railway  bridge  (p.  457)  is  128'  span,  and  24.25'  rise  — 
64*  =  24.25  xR 


To  find  the  depth  of  keystone,  by  rule  above,  as  in  one  of  a  series  — 
d  =   1/17x169  =    t/2ST73  =  5.33 

It  will  be  observed  that  the  depth  of  the  voussoir  or  arch-stone  is  in- 
creased in  most  bridges  from  the  keystone  to  the  springing  course,  but  not 
always,  nor  can  any  rule  be  deduced  from  the  examples  given.  It  is  safer 
so  to  increase  the  depth,  and  adds  but  little  to  the  expense—  increasing  the 
arch-masonry,  and  decreasing  the  spandrel-backing. 

Every  portion  of  an  arch  in  perfect  equilibrium  is  equally  pressed,  the 
lines  of  pressures  passing  through  the  centres  of  the  joints  of  the  several 
voussoirs  —  a  condition  seldom  found  in  practice,  but  compensated  by  an 
extra  depth  of  joint,  so  that  the  line  of  pressures  will  not  pass  without  the 
limits  of  the  joint.  If  an  arch  be  loaded  too  heavily  at  the  crown,  the  line 


ENGINEERING   DRAWING.  459 

of  pressures  passes  above  the  extrados  of  the  crown,  and  the  key  is  thrown 
down,  while  the  line  of  pressures  Mis  below  the  line  of  intrados  at  the 
haunches,  and  these  are  thrown  upward,  the  arch  separating  into  four 
pieces,  and  vice  versa  if  the  arches  are  overloaded  at  the  haunches.  To 
prevent  such  effects,  especially  from  moving  loads,  in  construction  the 
arches  are  loaded  with  masonry  and  earth,  that  the  constant  load  may 
be  in  such  excess  that  there  may  be  no  dangerous  loss  of  equilibrium  by 
accidental  changes  of  load. 

To  find  the  level  to  which  the  spandrel-backing  should  be  built  up, 
take  a  mean  proportional  between  the  radius  of  the  intrados  at  the  crown 
and  the  depth  of  the  arch-stone  (prop.  LIV.,  p.  67),  which  is  the  depth  of 
top  of  spandrel-backing  below  extrados  at  crown.  In  practice,  the  upper 
surface  of  the  spandrel-backing  is  never  level  ;  a  certain  amount  of  pitch  is 
necessary  to  discharge  the  moisture  which  may  percolate-  through  the 
earth-load  ;  but  the  rules  above  given  secure  plenty  of  load,  if  the  outer 
edge  be  dropped  somewhat,  and  the  inner  raised  a  little  toward  the  crown. 

To  determine  the  horizontal  thrust  of  an  arch,  divide  the  span  into 
four  equal  parts,  and  to  the  weight  of  one  of  the  central  parts  add  -^  of  its 
difference  from  the  weight  of  one  of  the  extreme  parts,  multiply  the  sum 
by  half  the  span,  and  divide  by  the  rise  of  the  arch. 

Thus  the  weight  of  one  of  the  middle  quarters  of  the  experimental 
arch  (table,  p.  457)  for  1  ft.  in  width  may  be  estimated  at  12,000  Ibs.,  an 
extreme  quarter  at  13,800  Ibs.,  the  quarter  of  the  distributed  load  of  300 
tons,  or  672,000  Ibs.,  divided  by  12,  or  the  width  in  feet  of  the  arch,  would 
give  14,208  Ibs.  for  the  ring  of  1  ft.  width  ;  the  sum  would  then  be 

=  26300 


6 
half  span  =  62  ft.  ;  rise  6.92 

=  235-634'r6>  horizontal  thrust  - 
To  determine  the  size  of  a  tie-rod  to  resist  this  thrust,  it  would  only  be 

necessary  to  divide  it  by  10,000  (p.  227)  ;  but  this 

thrust  is  to  be  resisted  by  the  masonry  of  the 

abutment  and  the  earth-load  behind  it. 

Thus,  if  fig.  46  be  a  section  of  an  abutment 

of  an  arch,  the  horizontal  thrust  exerted  at  T 

is  resisted  by  the  mass  of  masonry  of  the  abut- 

ment, the  tendency  is  to  slide  back  the  abutment 

on  its  base  A  C,  or  turn  it  over  on  the  point  A. 


The  sliding  motion  is  resisted  by  friction,  being  Fig.  46. 


460  ENGINEERING   DEAWESTG. 

a  percentage,  say  from  £  to  f ,  of  the  weight  of  the  abutment  and  of  half 
the  arch  which  is  supported  by  this  base ;  but,  in  turning  over  the  abut- 
ment on  the  point  A,  the  action  may  be  considered  that  of  a  lever,  the 
force  T  acting  with  a  lever  T  C  to  raise  the  weight  of  the  abutment  on 
a  lever  A  B  (G  being  the  centre  of  gravity,  and  G  B  the  perpendicular  let 
fall  on  the  base),  and  the  weight  of  half  of  the  arch  on  the  lever  A  C.  That 
is,  to  be  in  equilibrium,  the  horizontal  thrust  T  x  T  C  must  be  less  than  sum 
of  the  weights  of  the  abutment  multiplied  by  A  B,  and  the  weight  of  the 
arch  multiplied  by  A  C.  Seldom,  in  practice,  are  the  centres  struck,  and 
the  arch  allowed  to  exert  its  thrust,  till  the  abutment  is  backed  in  with 
earth,  well  rammed ;  and,  in  loading  the  arch,  great  care  is  taken  in  com- 
pacting the  earth  and  keeping  the  load  on  the  sides  balanced,  and  not  too 
much  at  the  sides  without  any  load  on  the  crown. 

SJcew  Iwidges  are  those  in  which  the  abutments  are  parallel,  but  not  at 
right  angles  to  each  other,  and  the  arches  oblique.  To  construct  these  in  cut 
stone  involves  considerable  intelligence,  both  in  the  designer  and  stone- 
cutter ;  but  when  the  work  is  laid  full  in  cement,  so  that  the  joints  are  as 
strong  as  the  material  kself,  this  refinement  of  stone-cutting  is  not  neces- 
sary. The  arch  may  safely  be  constructed  as  a  regular  cylinder  of  a  diam- 
eter equal  to  the  rectangular  distance  between  the  abutments,  with  its  ex- 
tremity cut  off  parallel  to  the  upper  line  of  road.  For  such  arches  hard- 
burnt  brick  is  the  best  material,  the  outer  voussoirs  being  cut  stone. 

CLASS  2. — Girder  or  Frame  Bridges,  resting  on  Piers. 

"Whatever  may  be  the  form  of  truss  or  arrangement  of  the  framing,  pro- 
viding that  its  weight  only  acts  on  the  abutment,  the  tension  of  the  lower 
chord,  or  the  compression  of  the  upper  chord,  at  centre,  may  be  determined 
by  this  common  rule : 

ffule. — The  sum  of  the  total  weight  of  the  truss,  and  the  maximum  dis- 
tributed load  which  it  will  be  called  on  to  bear,  multiplied  by  the  length  of 
the  span,  and  divided  by  8  times  the  depth  of  the  truss  in  the  middle,  the 
quotient  will  be  the  tension  of  lower  chord  and  compression  of  upper  at 
the  middle.  In  nearly  all  the  forms  of  diagonal  bracing,  if  the  uniform 
load  be  considered  as  acting  from  the  centre  toward  each  abutment,  each 
tie  or  brace  sustains  the  whole  weight  between  it  and  the  centre,  and  the 
strain  is  this  weight  multiplied  by  the  length  of  tie  or  brace,  divided  by 
its  height.  Any  diagonals,  equally  distant  from  the  centre,  sustain  all  the 
intermediate  load,  if  rods,  as  in  fig.  48,  by  tension ;  if  braces,  fig.  47,  by 
compression. 

It  follows,  therefore,  that  in  all  these  trusses  the  upper  and  lower  chords 


ENGINEERING   DRAWING. 


461 


should  be  stronger  at  the  centre  than  at  the  ends,  while  diagonals  should 
be  largest  at  the  abutments.  Unless  the  weight  of  the  bridge  is  great  com- 
pared with  the  moving  loads,  counter-braces  become  necessary  (p.  231). 


Y 


Fig.  49  is  a  side  elevation  of  one-half  the  gallows-frame  of  a  draw- 
bridge ;  fig.  50  an  elevation  and  part  cross-section  of  same ;  and  fig.  51 
plan  of  bottom  chord. 


fir        u 

Fig.  49. 


<S>O 


Fig.  50. 


462  ENGINEERING   DRAWING. 

The  whole  length  of  bridge  was  13SJ' — giving  two  clear  spaces  or 
water-ways  on  each  side  of  central  or  draw  pier,  a  little  exceeding  50  ft. 
The  gallows-frame  rested  on  a  turn-table  on  this  central  pier.  The  truss  is 
the  usual  form  of  Howe's  patent.  The  upright  posts  of  gallows-frame  are 
WxlO";  the  tension  rods  are  in  pairs,  2"  in  diam.,  one  on  each  side  of 
the  truss,  and  extending  from  the  top  of  frame  to  the  bottom  of  the  truss, 
at  a  distance  of  21  ft.  from  the  ends.  Fig.  51  explains  the  construction  of 
the  chords  of  this  bridge,  how  the  joints  are  broken,  and  how  the  timbers 
are  clamped  and  bolted  together. 

The  general  rule  adopted  in  the  construction  of  the  Howe  truss  is,  to 
make  the  height  of  the  truss  -J-  of  the  length  up  to  60  ft,  span ;  above  this 
span  the  trusses  are  21  ft.  high,  to  admit  of  a  system  of  lateral  bracing, 
with  plenty  of  head  clearance  for  a  person  standing  on  the  top  of  a  freight- 
car.  From  1T5  ft.  to  250  ft.  span,  height  of  truss  gradually  increased  up  to 
25  ft.  Moving  load  for  railroad  bridge  calculated  at  1  ton  per  running 
foot, 

Extract  from  specifications  for  (road)  bridge  on  site  of  Macomb's  Dam, 
New  York  city,  Howe's  truss : 

"  The  spans  at  the  ends  of  the  draw  shall  each  be  175'  long  in  the  clear,  resting  6' 
upon  the  piers  and  abutments,  on  suitable  bolsters.  The  height  of  truss  to  be  20'  from 
outside  to  outside  of  chords.  The  bottom  chords  will  each  be  formed  of  4  timbers, 
7£"  x  14"  each.  The  top  chords  each  of  4  pieces,  7J"  x  12".  The  main  braces  to  ave- 
rage 9"  x  10"  scantling.  Counter-braces  to  average  7"  x  9".  The  suspension-bolts,  or  tie 
rods,  of  which  there  shall  be  3  to  each  panel  of  truss,  to  be  1 1"  diam.  at  end  of  bridge, 
and  gradually  decrease  to  1 J"  diam.  at  centre.  The  roadway  to  be  20'  wide  between 
the  chords.  The  floor-timbers  to  be  80'  4"  x  10'',  placed  at  2'  from  centre  to  centre,  to 
alternately  project  4' outside  the  chords,  to  support  sidewalks.  The  space  between 
trusses  to  be  planked  with  3"  white-oak  plank;  sidewalks  with  2''  pine  plank." 


Fig.  52. 


Fig.  52  is  side  elevation,  plan,  and  section  of  cast-iron  girder,  adopted 
by  Mr.  Joseph  Cubitt,  C.  E.,  for  railway  purposes,  a  pair  of  girders  for 
each  track,  the  rails  being  supported  on  wooden  cross-beams. 


ENGINEERING   DRAWING. 


463 


Dimensions  for  different  Spans. 


Opening. 

Bearing  on 
abutment. 

Height  of  gird- 
er at  centre. 

Top  flanch. 

Bottom  flanch 
at  centre. 

. 
At  end. 

Thickness  of 
middle  web. 

12  ft. 

I'.G" 

I'A" 

3"  X  IV 

1'.4"  x  IV 

1'.8"    xlV 

IV 

30  ft. 

2'.G" 

3'. 

5"  X  2" 

I'.G"  x  2" 

I'.10"x2" 

2" 

45  ft. 

2'.9" 

• 

3'.9" 

7"  x  2V 

2'.      x  2V 

2'.        x  2V            2" 

The  upper  flanch  is  made  somewhat  stronger  than  Hodkinson's  rule,  to  withstand 
the  lateral  strain  or  vibration. 

Fig.  53  is  a  side  elevation  of  the  widest  span  wrought-iron  truss  over  the 
Connecticut  Eiver  on  the  N.-EL,  H.  &  S.  E.-E.,  designed  and  built  by 
Mr.  Laurie,  C.  E. : 

"This  girder  is  177'  long;  there  are  others  on  the  same  bridge  much  less,  but  the 
general  form  adopted  in  all  but  the  two  shortest  spans  is  that  of  a  truss  composed  of 


Fig.  53. 

rolled  plate  angle  and  T  iron.  There  are  three  distinct  varieties  of  this  general  form 
adopted  for  the  different  length  of  spans,  by  which  the  use  of  bars  beyond  a  certain 
size  is  avoided  in  the  longer  spans.  The  difference  consists  in  the  arrangement  of  the 
tie-bars.  In  the  span  of  177',  the  ties  cross  3  of  the  panels  formed  by  the  vertical  posts ; 
in  the  140'  and  88$-'  spans  they  cross  2  panels,  while  in  the  76£'  span  they  cross  but  one 
panel.  Where  the  ties  cross  3  panels  diagonally,  the  truss  partakes  somewhat  the  char- 
acter of  a  lattice,  and  the  principle  is  capable  of  being  ex- 
tended still  further  for  longer  spans,  by  making  the  tics 
cross  more  panels. 

"  The  top  and  bottom  chords,  of  which  a  section  is 
shown,  fig.  54,  are  composed  of  horizontal  plates,  26"  wide, 
varying  in  thickness.  At  right  angles  to  these  are  2  ver- 
tical plates,  placed  15"  apart,  and  connected  with  the  hor- 
izontal plates  by  4  angle  irons  in  each  chord,  to  which 
both  are  riveted.  The  horizontal  and  vertical  plates,  ex- 
cept at  the  ends  of  the  girders,  are  mostly  in  lengths  of 
15'  0",  the  joints  coming  between  the  posts  of  the  truss. 
At  the  joints  in  the  plates  there  are  covers  to  make 
strength  uniform.  Rivets  in  chords  1"  diam.,  8-}£"  apart. 
Batween  the  lower  edges  of  the  vertical  plates  of  the 
upper  chords  there  are  wrought-iron  distance  pieces,  one 


Fig.  54. 


46-i  ENGINEERING   DRAWING. 

to  each  panel,  by  which  the  two  plates  are  held  securely  in  place,  and  give  the  chords 
additional  stiffness.  The  top  chord  at  the  middle  has  2  horizontal  plates,  26"  x  f " ;  4  angle 
irons,  4"  x  4"  x  £",  and  two  vertical  plates,  15"  x  f  ",  making  a  sectional  area  of  76.2  sq.  in. ; 
at  the  end  there  is  one  horizontal  plate,  26"  x  f  " ;  angle  irons,  4"  x  4"  x  V ;  vertical  plate, 
15"  x  £".  The  bottom  chord  at  centre,  horizontal  plates,  26"  x  f",  and  26"  x  f " ;  at  ends 
one  plate,  26"  x£".  Length  of  girder,  177'.3",  in  panels  of  5'.3";  height  16'.9"  between 
horizontal  plates.  Width  between  centres  of  girders,  10'.6".  End-posts  are  composed 
of  6  T  bars,  5"  x  3$-"  x  \",  in  pairs,  with  2  side  plates,  25$"  x  £" ;  1  end  plate,  12V  *  f". 
The  next  post  has  2  T  bars,  5"  x  3V  x  \"  •  2  side  plates,  10"  x  ^". 

Posts    3  to    5  each  2  T  bars,       .        .        .        .        .        .  6"  x  4"   x  f  " 

6  to    8     "     2       " G"  x  4"    x  V 

"       9  to  10     "     2      '  5"x3VxV 

"     10  to  centre,    2      " 5"  x  3V  x  Ty 

"  All  the  posts  have  diagonal  bracing  between  the  T  bars ;  they  are  divided  into  5 
spaces  between  the  chords  by  cross-plates,  5"  x  T7ff",  with  diagonals,  2V  x  J".  Between 
the  vertical  plates  of  the  chords  the  T  irons  are  connected  by  plates  13"xl2"xV. 
The  posts  are  placed  between  and  riveted  to  the  vertical  plates  of  the  chords.  Xear 
the  ends  of  the  truss  there  are  10  rivets  on  each  side,  top  and  bottom;  2  being  through 
the  angle  irons,  which  connect  the  vertical  and  horizontal  plates,  the  others  through 
the  vertical  plates.  The  number  is  diminished  toward  the  centre,  according  to  the  sec- 
tional area  of  the  post.  Rivets  through  posts  and  chord,  1"  diam.,  in  diagonal  bracing, 
f  "  diam. 

"  The  ties  are  in  pairs :  the  1st  from  end  crosses  1  panel ;  the  2d  2 ;  the  3d  3.  The 
ties  extend  2  panels  beyond  the  middle,  shown,  by  calculation,  to  be  all  the  counter- 
bracing  necessary.  The  ties  vary  in  width  from  8"  at  the  end  of  bridge  to  2V  at  centre, 
uniform  thickness,  f ".  The  ties  are  riveted  to  the  outside  of  the  vertical  plates  of  the 
chords,  part  of  the  rivets  also  passing  through  the  posts  inside  the  vertical  plates.  The 
number  of  rivets  in  each  bar  is  so  arranged  as  to  make  the  sectional  area  of  the  rivets 
fully  equal  to  that  of  the  bar,  and  so  placed  that  the  effective  area  of  the  tie-bar  is  only 
diminished  by  the  amount  of  metal  taken  out  by  one  rivet-hole. 

"  The  horizontal  bracing  across  the  top  and  bottom  of  the  two  trusses  is  formed  of 
T  bars,  placed  at  right  angles  to  the  girders,  at  intervals  of  10'  6",  or  1  on  every  2d 
post,  varying  in  size  from  6"  x  4"  x  V,  at  ends,  to  4"  x  4"  x  f"  at  middle.  Between  these 
are  horizontal  diagonals  of  round  iron,  varying  from  iy  to  iy  diam.  The  vertical 
diagonal  tie-rods  are  iy  diam.  at  ends,  the  rest  IV  diam.  Both  horizontal  and  verti- 
cal tie-rods  are  fitted  with  nuts  and  screws  for  tightening  when  necessary.  The  ends 
of  the  girders  rest  on  cast-iron  plates ;  one  end  is  firmly  fixed  to  pier,  the  other  resting 
on  rollers. 

"  The  superstructure  of  the  bridge  is  .formed  of  wooden  floor-beams,  17'  9"  long, 
7"  x  12"  laid  across  the  top  of  the  girders,  20"  from  centre  to  centre.  Upon  these  rest 
the.  longitudinal  stringers,  9"  x  15",  which  support  the  track." — Desertion  of  the  Iron 
Bridge,  etc.,  T.  G.  ELLIS,  C.  E. 

For  strength  of  \vrought-iron  box-girders,  and  the  comparative  strength 
of  different  spans  and  dimensions,  see  p.  127. 


ENGINEERING   DRAWING. 


465 


CLASS  3.— In  suspension  bridges  the  platform  of  the  bridge  is  sus- 
pended from  a  cable,  or  chains,  the  ends  of  which  are  securely  anchored 
within  the  natural  or  artificial  abutments.  From  the  nature  of  the  struc- 
ture, the  bridge  accommodates  itself  to  each  change  in  the  load,  assuming 
the  position  of  equilibrium  for  each  particular  load  to  which  it  is  tem- 
porarily subjected. 

The  curve  of  a  suspended  chain  is  that  known  as  the  Catenary,  and,  if 
the  whole  weight  of  the  structure  were  in  the  chain  itself,  this  would  be 
the  curve  of  the  chains  of  a  suspension  bridge ;  but,  as  a  large  part  of  the 
weight  and  the  whole  of  the  loading  lies  in  the  platform,  the  curve  assimi- 
lates to  that  of  a  parabola,  and,  in  all  calculations,  it  is  so  regarded. 

Let  fig.  55  represent  a  suspension  bridge,  with  the  roadway  or  platform 
F  L,  and  A  D  B  C  being  the  curve  formed  by  the  chain. 


Fig.  53. 

To  determine  the  form  of  the  chain,  calculating  the  position  of  the 
point  in  reference  to  the  horizontal  line  F  L :  the  data  for  this  are  the  semi- 
span  A  E,  the  deflection  E  B,  and  the  length  of  the  shortest  suspension 
rod  B  II. 

To  find  the  length  of  any  suspension  rod,  viz.,  D  G,  subtract  the  length 
of  the  shortest  suspension  rod  *B  II  from  the  deflection  E  B,  multiply  the 
remainder  by  the  square  of  the  horizontal  distance  D  K,  and  divide  by  the 
square  of  the  semi-span  A  E ;  to  the  quotient  add  the  length  of  the  short- 
est rod  B  II,  and  it  will  give  the  length  of  the  suspension  rod  D  G.  In 
the  same  way,  any  number  of  points  in  the  curve  may  be  determined, 
through  which  the  curve  can  be  determined. 

For  the  strain  of  tension  on  the  chain  at  the  points  of  support  A 
andC: 

Rule. — Add  together  four^imcs  the  square  of  the  deflection  (E  B)3  and 
the  square  of  half-span  (A  E)2,  and  take  the  square  root  of  this  sum ;  mul- 
tiply this  result  by  the  total  weight  of  one  chain  and  all  that  is  suspended 
from  it,  including  the  distributed  load,  and  divide  this  product  by  four 
times  the  deflection  (E  B)  of  the  cable  at  the  centre,  and  the  result  will  be 
30 


466 


ENGINEERING   DRAWING. 


the  tension  on  one  chain,  at  each  point  of  support,  A.  and  C.  The  angle 
made  by  the  chain  at  the  point  of  support,  viz.,  angle  POL  and  the  angle 
of  the  backstays,  or  continuation  of  the  chain  (angle  L  C  ^s")  should  be 
equal  to  each  other,  in  order  that  there  be  no  tendency  to  overset  the  tower 
C  L  and  A  F. 

The  horizontal  pull  in  the  direction  between  A  and  C,  if  the  chain  v:ere 
fastened  there,  would  be  the  tension  found  by  the  first  rule,  multiplied  by 
the  cosines  of  the  angle  P  C  E,  or  the  tension  at  the  point  of  support,  mul- 
tiplied by  E  0,  and  divided  by  C  P.  Therefore,  if  the  main  chains  and 
backstays  make  unequal  angles  with  the  tower,  the  difference  of  the 
cosines  of  these  angles  will  be  the  tendency  of  the  towers  to  overset. 

Having  determined  the  strain  on  the  chains,  it  is  easy  to  estimate  the 
size  necessary  to  resist  it ;  but,  when  the  chain  is  composed  of  links,  the 
size  of  pins  and  eyes  is  to  be  in  some  proportion  to  the  body  of  the  links. 
By  experiment,  Sir  Charles  Fox  established  a  rule,  that  the  diameter  of  the 
pin  should  be  -f  the  width  of  the  body  of  the  link,  and  the  width  of  the 
two  sides  of  the  eyes  should  be  about  10^  greater  than  the  body  of  the 
link,  link  and  eye  being  of  uniform  thickness  throughout. 


Main 
spans. 

Deflection 
of  chain  or 
cable. 

No.  of 

chains  and 
cables. 

Total  effective       Mean  weight     Fixed  load 
section  of  cable  of  cable  per  ft.  of     per  ft.  of 
in  eq.  inches.          span  (\bs.).       span  (Ibs.). 

Breadth 
of  platform 
in  feet. 

Menai  

! 
570        43 

16 

260 

880 

28 

Chelsea.  .  .  . 

348        29 

4 

230 

767 

47 

Pesth  

666 

47.6 

4 

507 

1690 

9892 

46 

Bamberg.  .  . 

211 

14.1 

4 

40.2 

137 

1581 

30.5 

Freyburg  .  . 

870 

63 

4 

49 

167 

760 

21.25 

Niagara 
Falls  

821 

54  and  64 

4 

241.6 

820 

2032 

24 

Cincinnati.  . 

1057 

89        1        2 

172.6   ' 

516 

2580     1    36 

Steam-Engines. — Under  the  head  of  "  Mechanics,"  pp.  133-136,  rules 
are  given  for  determining  the  effective  power  of  a  steam-engine,  the  vol- 
ume of  steam  required  under  different  pressures  or  tensions  to  supply  this 
power,  the  quantity  of  water  required  for  this  volume  of  steam,  the  prob- 
able amount  of  coal  to  produce  the  evaporation  of  this  quantity  of  water, 
and  the  general  relative  proportions  of  grate  and  heating  surface  for  a 
boiler  to  do  this  economically.  It  will  be  observed  that  the  effective  power 
of  an  engine  is  measured  by  the  area  of  the  piston,  in  square  inches,  mul- 
tiplied by  the  average  pressure  of  steam,  in  pounds  per  square  inch  on  it. 
multiplied  again  by  its  travel,  in  feet  per  minute,  and  the  product  iV 
the  effective  power  in  Ibs.  ft  per  minute ;  or,  if  the  product  be  divided 


ENGINEERING-   DRAWING.  467 

by  33,000,  the  result  is  the  horse-power  (HP)  of  the  engine.  Hence,  with 
the  very  same  engine,  if  either  the  steam-pressure  or  travel  of  piston 
be  varied,  the  power  of  the  engine  is  changed ;  thus,  an  engine  of  10" 
diameter  (A  =  78.54  square  inches),  2  ft.  stroke,  with  an  average  pressure 
of  30  Ibs.  per  square  inch,  making  50  revolutions  per  minute  (travel, 
50x2x2),  would  give  nearly  14f  HP — 

78.54  x  30  x  200^33,000  =  145  ; 

but  if  the  average  pressure  were  60  Ibs.,  or  the  travel  400  ft.,  the  HP  would 
be  double ;  or,  if  pressure  and  travel  be  both  doubled,  the  HP  would  be 
quadrupled;  and,  within  the  limits  of  practice,  this  same  engine  would 
give  an  actual  100  HP,  provided  there-were  a  boiler  of  sufficient  capacity 
to  supply  the  steam.  It  is  evident,  therefore,  that  when  mechanics  speak  of 
an  engine  as  so  many  horse-power,  it  is  merely  a  conventional  term,  mean- 
ing a  certain  size  in  the  idiom  of  a  particular  shop,  but  has  no  general 
acceptation  which  defines  the  dimensions.  Among  the  mechanics  them- 
selves it  is  very  common  to  give  the  size  in  inches  diameter  by  inches  stroke ; 
thus,  in  engine  above,  10"  x  24",  the  diameter  is  given  first;  but  to  their  cus- 
tomers they  speak  of  so  many  horse-power.  The  term  was  first  introduced 
by  Watt,  who  made  experiments  on  the  strength  of  horses,  and  rated  his  en- 
gines at  so  many  nominal  HP  by  fixed  rules.  Since  his  time,  his  standard 
of  pressure  and  travel  has  been  very  much  increased,  and  the  engine  de- 
livered for  so  many  nominal  HP,  according  to  Watt,  will  give  6  to  12 
times  as  many  actual  or  effective  HP. 

In  the  same  way  it  is  very  usual  to  speak  of  boilers  as  so  many  HP, 
meaning,  probably,  that  they  have  the  capacity  of  evaporation  to  supply 
steam  for  so  many  HP ;  yet  there  is  no  standard  of  volume  of  steam  or 
water  to  supply  a  IIP.  Under  various  conditions  of  cut-off  and  condensa- 
tion, one  engine  will  require  -|  the  steam  of  another  to  develop  the  same 
power ;  and,  under  the  varying  conditions  of  fuel  and  draught,  the  same 
boilers  will  evaporate  twice  as  much  water,  with  but  little  difference  in 
economy  of  combustion ;  and  different  types  of  boilers,  with  equal  and 
well-proportioned  grates  and  heating  surfaces,  will  consume  very  different 
amounts  of  coal,  and  evaporate  very  different  quantities  of  water. 

"  Thus,  the  Cornish  boilers  at  Jersey  City  "Water  "Works,  Belleville,  N.  J.,  are  7  ft. 
diameter  x  84  ft.  long ;  grate-surface,  20.22  sq.  ft. ;  single  central  flue,  4'  4"  diameter, 
through  which  the  products  pass  to  the  end  of  the  boilers,  and  then  return  by  wheel- 
draught  at  the  sides,  and  back  beneath  boiler.  Cumberland  coal  burned  per  square  foot 
of  grate  was  4,839  Ibs.  per  hour,  and  evaporation  10.50  Ibs.  of  water  per  pound  of 
coal. 

"  The  drop-flue  boiler  (fig.  5G)  at  the  Hartford  Water  Works  is  7'  6"  diameter,  22'  8" 


468 


ENGINEERING   DRAWING. 


long;  four  direct  flues,  18"  diameter,  14'  4"  long;  1  return-flue,  12"  diameter;  2  9" 
and  13  8" — all  12'  3"  long ;  grate,  23£  sq.  ft.  The  products  of.  combustion  pass  through 
the  18"-flues  to  the  back-connections,  thence,  dropping  down,  pass  through  the  smaller 


Fig.  56. 

return-flues  to  the  front-connection,  thence  back  beneath  the  boiler  to  the  chirnney-flue. 
Cumberland  coal  burned  per  square  foot  of  grate,  5.683  Ibs. ;  evaporation,  10.96  Ibs.  of 
water  per  pound  of  coal. 

"  Two  tubular  boilers,  at  the  Nashua  Manufacturing  Company's  mills,  Nashua,  N.  H., 
each  5'  diameter,  20'  long;  55  tubes,  3V'  diameter;  grate  beneath  boiler-shell,  4'  loy  x 
5'  6£".  The  products  of  combustion  pass  beneath  the  boiler,  return  through  the  tubes, 
thence  back  through  the  flues  of  a  heater  placed  above  and  between  the  boilers ;  fire- 
surface  in  heater,  160  sq.  ft.  Coal,  anthracite,  burned  per  square  foot  of  grate,  5.309 
Ibs. ;  evaporation,  9.18  Ibs.  of  water  per  pound  of  coal. 

"  Upright  cylindrical  boiler,  in  Massachusetts  Cotton  Mills,  Lowell,  Mass.,  31J" 
diameter,  12'  high,  3"  water-space  whole  height  of  boiler ;  pot  or  drum  in  centre  19" 
diameter,  9'  3"  high,  with  flue,  liy  diameter,  through  the  same ;  grate,  3.14  sq.  ft. 
Coal,  anthracite,  burned  per  square  foot  of  grate,  13.4  Ibs. ;  evaporation,  8.65  Ibs.  water 
per  pound  of  coal. 

"  Locomotive  boiler,  at  the  Boston  Cotton  Mills,  Lowell,  Mass. — The  products  of 
combustion  pass  from  the  fire-box  through  64  tubes,  2f"  diameter,  13'  9"  long,  to  a 
smoke-box  surrounded  by  water,  thence  under  the  boiler  through  a  heater ;  grate-sur- 
face, 16.66  sq.  ft. ;  heating  surface,  including  heater,  748  sq.  ft.  Anthracite  coal  con- 
sumed per  square  foot  of  grate,  7.52  Ibs. ;  evaporation,  8.48  Ibs.  of  water  per  pound 
of  coal. 

"  The  temperature  of  feed-water  reduced  to  the  standard  of  100°.  The  first  two 
experiments  conducted  by  Messrs.  Copeland  and  Worthen,  the  latter  by  Mr.  Jas.  B. 
Francis." 

In  the  above  experiments,  although  the  boilers  were  but  doing  their 
usual  work,  if  we  except  the  upright  boiler,  the  consumption  of  coal  per 
square  foot  of  grate  is  much  less  than  in  practice.  At  the  Ridgewood 
Pumping  Engines,  where  there  are  drop-flues  similar  to  those  at  Hartford, 
the  usual  consumption  of  coal  is  about  13  Ibs.  per  hour  per  square  foot  of 
grate,  but  with  somewhat  less  pounds  of  evaporation  than  in  the  above  ex- 


ENGINEERING-   DRAWING. 


469 


periment.  In  general  too  little  attention  is  paid  to  the  economy  of  large 
boiler-capacity,  and  it  may  be  safe  to  estimate,  for  the  general  boilers  in  use, 
an  evaporation  of  8  Ibs.  of  water  per  pound  of  coal,  and  a  combustion  of  12 
Ibs.  of  coal  per  square  foot  of  grate ;  and,  for  high-pressure  engines,  a  con- 
sumption of  5  Ibs.  of  coal  per  IIP  per  hour ;  steam-space  in  boilers  nearly 
equal  to  that  of  water-space. 

The  boilers  of  steamships  and  locomotives,  from  want  of  space,  have 
necessarily  the  heating  surface  very  much  concentrated,  and  an  artificial 
draught  to  increase  the  combustion. 

The  following  is  a  table  of  English  experience,  from  Rankin,  and  gives 
the  comparative  rate  under  different  boilers : 


Per  sq.  ft.  of 
grate  per  hour. 

4  Ibs. 
10     " 

.12  to    16     " 
.10  "     24     " 


With  Chimney-draught. 

Slowest  rate  of  combustion  in  Cornish  boilers 

Ordinary      "  "  "         "  "      

"  "  "  "    factory       "      

"  "  "    marine       "      

With  draught  produced  by  blast-pipe  or  fan. 
Locomotives 40  "  120    " 

In  deciding  upon  the  size  of  engine  and  boilers  to  do  a  certain  w^ork, 
there  is  to  be  determined  the  amount  of  work  in  Ibs.  ft.,  or  HP;  the  Id  ml 
of  work,  whether  the  resistances  are  uniform,  and,  if  so,  what  limits  the  pis- 
ton travel,  as  water  in  a  pumping-engine,  or  the  proper  speed  of  shaft  for 
a  mill,  or  for  a  propeller ;  or,  if  the  resistance  be  not  uniform,  as  in  an 
iron-rolling  mill,  what'  velocity  may  be  required  in  moving  parts  to  over- 
come the  maximum  resistance ;  what  is  the  space  that  can  be  occupied  for 
the  purposes  of  power ;  whether  the  engine  is  to  be  condensing  or  not ; 
low  or  high  pressure ;  and  whether  economy  in  first  cost  is  more  desirable 
than  in  maintenance. 

The  limits  of  ordinary  travel  of  stationary  engine-pistons  is  from  200' 
to  500'  per  minute ;  average  pressure  in  low-pressure  condensing  engines, 
from  10  to  15  Ibs. ;  boiler-pressure,  from  15  to  25  Ibs.  Average  pressure 
of  high-pressure  stationary  engines,  from  25  to  50  Ibs. ;  boilers,  45  to  75  Ibs. 

"  Fairbairn  gives  the  following  table  of  the  safe  working-pressure  of  boilers  of  dif- 
ferent diameters,  44$  being  allowed  for  loss  of  strength  by  rivet-holes : 


Diam. 
3 

%"-plate. 
118 

157  25 

Diam. 
6.0    

%"-plate. 
....     59 

78.75 

o  6 

101 

13475 

6.6... 

.  54.25 

72.5 

4  0 

885 

118 

7.0  

50.50 

67.25 

4  6 

7875 

104.75 

7.6  

47. 

62.75 

5 

70.75 

94.25 

8  

44. 

59. 

5.6... 

.  64.75 

85.75 

8.6... 

.  .  41.5 

55.5 

470 


ENGINEERING   DRAWIXG. 


"According  to  Holley  ('Railway  Practice'),  the  ordinary  American  plate  is  4 
stronger  than  that  of  the  English,  and  for  a  48"-shell  it  is  not  uncommon  here  to  use 
plates  of  from  £"  to  Ty  under  a  working-pressure  of  120  Ibs.  The  thickness  of  fire- 
plates  does  not  seem  to  add  to  their  durability.  J"  Lowrnoor  are  successfully  used 
here,  while  V  Lowmoor  fail  after  a  few  months'  use  in  England.  Joints  of  a  double 
thickness  of  metal,  and  rivet-heads  in  a  fire-box,  give  way  sooner  than  the  single  plate. 

"  Rivets  are  commonly  from  £ "  to  £"  diameter,  and  pitched  at  If"  to  2"  centres. 
The  maximum  strength  is  obtained  when  the  sectional  area  of  the  rivets  is  £  that  of  the 
punched  plates ;  lap  of  plates,  about  2". 

u  Mr.  Fairbairn  made  experiments  on  the  strength  of  flues  or  tubes,  which  he  found 
to  be  inversely  as  their  diameters  and  their  lengths,  and  directly  as  the  2.19-power  of 

fjv;2.10 

the  thickness  of  the  plates ;  collapsing  pressure  in  Ibs.  =  806,300  ;  and  gives  the 

Li  J_) 

following  table  of  equal  strength  of  cylindrical  flues  for  a  collapsing  pressure  of  450 
Ibs.  per  square  inch 


Biain. 

12" 

Length  of  flues. 
10'        20'         30' 
Thickness  of  plate. 
291"      399"      480" 

Diam. 
86" 

Length  of  flues. 
10'        20'         30' 
Thickness  of  plate. 
480"      659"      794' 

18 

350       480       578 

42  . 

516        707        Sol 

01 

399       548       659 

48  . 

.548       752       905 

30   .-. 

.  .442      .607      .730 

These  calculations  are  based  on  the  supposition  that  the  flues  are  plain 
cylinders  ;  but  it  is  now  the  practice  in  England  to  make  the  joints  as  in 

figs.  57  and  58.    The  circular  joints 
are  made  with  T  or  fl  iron,  which, 
in  effect,  is  virtually  shortening  the 
Fis-57-  ms-58.  tubes.      The  T  and  U  are  in  the 

water-space  ;  a  space  is  left  between  the  sheets  in  the  flue  for  calking ;  the 
horizontal  joints  are  butt,  with  a  single  welt. 

In  boiler  or  other  plate-work,  where  two 
joints  at  right  angles  to  each  other  are  butt- 
joints  covered  with  a  welt,  the  intersecting  welts 
are  made  by  scarfing  the  one  and  chamfering  the 
other,  as  shown  in  section  at  b,  fig.  59. 

Plate  CXYI.  is  longitudinal  (fig.  1)  and  half- 
\  transverse  (fig.  2)  sections  of  the  fire-box  of  an 
anthracite -burning   locomotive   from  the  !Kew- 
Jersey  R.-Tt.,  and  illustrates  very  fully  the  differ- 
ent kinds  of  stays  in  use  in  such  construction. 
The  water-space  is  4"  wide  in  front,  3"  on 
In  this  example  the     in- 


o    o 


Fig.  59. 


sides,  and  6"  behind. 


ENGINEERING   DRAWING.  -471 

closing  plates  are  parallel;  but  it  is  considered  a  very  good  practice  to 
make  the  inside  plate  overhang  a  little,  giving  a  wider  water-space  at 
top  than  bottom.  The  stay-bolts  in  water-space  are  |"  diameter,  and 
4"  centres ;  they  are  screwed  into  inside,  and  riveted.  It  is  common  to 
make  these  bolts  of  tubes,  fastening  them  the  same  as  the  tubes  in  the 
boiler,  and  closing  up  the  end  with  an  iron  plug,  except  where  it  may 
be  convenient  to  introduce  air  through  them  into  the  fire-box.  The  bot- 
tom of  the  water-space  is  made  with  wrought-iron  ring,  the  inside  plate  be- 
ing bent  down  a  little,  as  shown,  and  the  whole  riveted  strongly  together. 
The  opening  for  the  door  is  made  by  turning  a  fianche  on  the  inside  plate, 
to  which  a  plate-ring  is  riveted,  which  is  also  riveted  to  an  angle-iron  ring, 
riveted  to  the  outside  plate.  The  crown-sheet  of  fire-box  is  supported  by 
cast-iron  girders,  extending  across  the  boiler ;  these  girders  are  cast  double, 
with  a  space  for  the  insertion  of  the  bolts  ;  the  ends  of  these  girders  rest 
on  the  inner  plates  of  the  fire-box,  but  they  are  also  supported  by  hangers 
(h  h)  from  the  outer  shell,  and  the  inside  of  the  steam-drum,  being  mutual 
stays  for  the  crown  of  the  fire-box  and  roof  of  boiler.  These  hangers  have 
a  fork  at  one  end,  through  which  a  pin  is  passed  to  connect  it  with  the  foot 
riveted  to  the  boiler ;  the  other  end  passes  into  the  space  in  the  girder,  and 
a  pin  is  passed  through  girder  and  hanger.  It  will  be  observed  that  there 
are  stays  (s  s)  for  the  boiler-front,  extending  back  to  inclined  part  of  the 
shell,  and  a  similar  one  (s)  in  the  angle  of  the  fire-box  beneath  the  tubes. 
For  the  staying  of  the  fiat  surface  of  boiler-fronts, 
stays  like  the  hangers  are  often  used,  at  an  angle  con- 
necting the  end  with  upper  shell  or  triangular  plates 
(fig.  60),  called  gussets,  riveted  in  angle  of  shell.  The 
boiler  has  130  2//-tubes,  and  26  S^'-tubes;  space  be- 
tween tubes,  -g" ;  length,  10  ft. 

Plate  CXYII.  are  drawings  of  a  tubular  boiler, 
with  grate  beneath.  Fig.  1  is  a  side  elevation  of  the 
boiler,  with  walls  and  front  in  section.  The  boiler  is  Fis- 60- 

represented  broken,  as  the  page  would  not,  on  this  scale,  admit  the  full 
length. 

Fig.  2  is  a  half-front  elevation,  and  half  section  through  grate  and  front 
flue.  B  is  the  boiler ;  D  the  steam-drum  ;  v  is  the  safety-valve  ;  s  the  pipe 
for  steam-connection ;  b  for  steam  blow-ofi",  or  waste  from  safety-valve ;  m 
the  man-hole,  and  h  the  hand-hole  ;  S  the  saddle  supporting  the  rear  end 
of  boiler ;  the  front  is  supported  on  frame,  cast  with  the  front  f.  There 
are  also  3  brackets  riveted  on  each  side  of  boiler,  and  resting  on  wall ;  g 
are  the  grates,  d  the  fire-door,  and  d'  the  ash-pit  door,  w  the  bridge-wall ; 


472  ENGINEERING    DRAWING. 

p  is  the  feed-pipe  for  supplying  water  to  the  boiler.  By  an  arrangement 
of  valve  the  same  pipe  may  be  also  used  as  a  blow-off,  or  to  draw  off  the 
water  from  the  boiler.  The  products  of  combustion  arising  from  the  grate 
pass  over  the  bridge  and  around  the  lower  hemisphere  of  the  boiler  to  its 
rear,  thence  through  the  tubes  to  the  front  flue,  and  thence  by  the  side  flue 
f '  into  the  chimney.  Access  is  had  to  the  flue  f,'  and  the  tubes,  through 
the  door  c  hung  on  the  front.  It  will  be  observed  that  a  part  of  the 
boiler  projects  over  the  flue  f,'  forming  what  is  called  by  mechanics  a  hog- 
nose;  but  by  many  this  is  omitted,  and  only  the  shell  of  the  boiler  projects 
the  entire  circumference, with  a  pipe  shown  by  the  dotted  line,  connecting 
with  a  larger  horizontal  circular  flue,  passing  over  and  receiving  like  pipes 
from  other  boilers  in  the  row.  To  protect  the  boiler  from  loss  of  heat  by 
radiation,  it  is  the  most  common  practice  to  cover  the  upper  half  of  the 
shell  with  ashes  to  the  depth  of  from  V  to  IS",  and  the  steam-drum  by 
felting.  The  side-walls  by  the  grate,  the  bridge-wall,  the  front,  and  all 
parts  exposed  to  the  direct  action  of  the  fire,  are  lined  with  fire-brick. 

Many  boilers,  like  locomotives,  are  not  set  in  brickwork ;  these  it  is 
usual  to  cover  with  felting.  The  protection  of  all  parts  of  boilers  and 
steam-pipes  exposed  to  the  air  by  some  cover  of  a  non-conducting  ma- 
terial adds  much  to  economy  in  the  consumption  of  coal,  and  dryness  of 
steam. 

Plate  CXVIII.  are  sections  of  two  chimneys.  Fig.  1  is  a  section  of  a 
chimney  attached  to  an  English  gas-house,  taken  from  "Engineering," 
and  fig.  2  a  section  of  the  chimney  at  the  Ridgewood  Pumping-eugine 
House. 

Fig.  1  is  given  as  an  example  of  a  very  neat  chimney,  uniform  flue  and 
shell,  additional  strength  being  given  by  the  buttresses  shown  in  section, 
fig.  3.  It  differs  from  the  chimneys  usually  constructed,  in  having  no  in- 
dependent flue  inside,  as  shown  in  the  section  of  the  Ridgewood  chimney, 
which  can  freely  expand  with  the  heat  without  affecting  the  outer  shell. 
Fig.  4  is  an  elevation  of  the  Ridge  wood  chimney,  at  the  point  where  the 
square  base  is  changed  into  an  octagonal.  Fig.  5  is  a  section  of  the  shaft, 
but  the  flue  should  have  been  represented  circular. 

For  the  area  of  chimney-flues  one  square  inch  for  every  pound  of  coal 
burnt  per  hour  on  the  grate  has  been  found  to  answer  well  in  practice. 
Chimneys  are  constructed  of  various  sections,  sometimes  uniform  through- 
out their  length,  sometimes  tapering  at  the  top,  and  sometimes  bell- 
mouthed  ;  all  answer  the  purpose.  The  great  point  to  be  observed  is,  that 
there  be  no  abrupt  changes  of  section  or  direction,  and  that  they  be  carried 
well  above  all  disturbing  causes. 


ENGINEERING  DRAWING. 


473 


Plate  CXIX.  contains  a  side-elevation  and  some  details  of  a  simple 
form  of  stationary  engine.  Fig.  1  is  the  elevation  in  which  S  is  the  steam- 
cylinder,  p  the  piston-rod  attached  at  its  outer  end  to  a  cross-head  h,  slid- 
ing on  the  guides  g  ;  the  connecting  rod  r  connects  the  cross-head  and 
crank  c  on  the  fly-wheel  shaft.  The  fly-wheel  is  only  shown  in  part  ;  on 
the  same  shaft  is  an  eccentric  giving  motion  to  the  valve-rod  r,  and  a  pulley 
to  drive  the  governor  with  not  shown  on  drawing.  The  rod  e  hooks  on 
to  a  rocker,  r',  to  which  is  attached  the  valve-rod  by  means  of  the  handle 
h'  ;  the  rod  e  can  be  unhooked  from  the  rocker,  and  the  valves  moved  by 
hand.  The  steam  is  introduced  through  the  steam-pipe  s  beneath  the 
cylinder,  and  exhausted  through  the  pipe  E.  The  cast-iron  engine-frame 
F  rests  upon  a  stone  base,  B,  on  a  brick  or  stone  foundation,  to  which  it  is 
strongly  bolted,  and  which  is  shown  separately  (plate  CXX.). 

Fig.  2  is  a  section  of  the  cylinder  ;  P  the  piston,  C  the  steam-chest,  v  v 
the  valves,  S  the  steam-connection,  and  E  the  exhaust,  Access  may  be  had 
to  the  valves  by  taking  off  the  bonnets  b  b  ;  b'  is  the  stuffing-box.  Bolts 
are  not  shown  on  the  drawing,  but  the  different  pieces  will  be  understood 
by  the  cross-hatching.  The  valve-chest  is  bolted  on  face  of  cylinder,  of 
which  fig.  3  is  a  plan  of  a  part  showing  p'  and 
p",  the  steam  and  exhaust  parts,  and  the  bolt- 
holes  tapped  in  face  to  receive  valve-chest  bolts. 

Fig.  61  is  section  and  plan  of  stuffing-box  for 
steam  piston-rod.  It  consists  of  two  parts  —  the 
box  b,  which  is  attached  to  the  cylinder-head, 
and  is  bored  out  somewhat  bigger  than  the  pis- 
ton-rod, except  at  the  bottom  ;  and  the  gland  g, 
which  is  turned  to  fit  the  box,  and  bored  to  fit 
the  piston.  The  space  in  the  box  is  filled  by  a 
gasket,  or  other  packing,  and  the  gland  is  then 
forced  in  by  the  screws,  compressing  the  pack- 
ing and  making  a  tight  fit  around  the  piston- 
rod.  The  dimensions  vary  with  the  diameter 
of  piston-rod,  depth  of  box  being  from  3"  to 
12",  and  space  for  packing  £"  to  1".  In  small 
engines,  the  gland  is  often  screwed  into  the 
box,  a  follower  or  ring  being  placed  above  the 
packing. 

Fig.  62  is  a  sectional  plan,  and  fig.  63  a  sectional  elevation,  of  a  part 
of  the  exterior  of  a  piston,  showing  the  common  form  of  ring-packing, 
which  consists  of  a  single  interior  ring  (r),  and  two  exterior  rings  (r"  r"), 


Fig.  ei. 


EXGDfEEKLNXJ   DEAWLN'G. 


each  cut  in  two,  and  so  fastened  that  the  joints  are  always  broken.  The 
packing  is  set  out  by  springs,  one  of  which  is  shown  at  s.  F  is  the  fol- 
lower, which  can  be  taken  oif  for  the  admission  of  the  rings,  and  then 


Fig.  62. 


Fig.  63. 


replaced  and  bolted  to  piston,  making  a  close  joint  with  end  of  rings. 
The  dimensions  vary  with  the  diameter  of  cylinder,  the  thickness  of  the 
piston  being  from  3"  to  9"  at  exterior,  with  a  swell  in  the  centre  for  large 
pistons. 

Plate  CXX. — Figs.  1,  2,  and  3  are  side  and  end  elevation,  and  plan, 
of  the  foundation  of  the  stationary  steam-engine  (pi.  CXIX.).  F  is  the  cast- 
iron  frame  or  bed-plate  of  the  engine ;  B  the  granite  bed  of  engine,  or 
coping  of  foundation  ;  P  the  stone  or  brick  pier,  laid  full  in  cement.  The 
granite  bed  is  levelled  accurately,  and  well  hammered,  to  receive  the  engine- 
frame.  Strong  wrought-iron  bolts  pass  through  frame,  bed,  and  pier,  with 
nuts  at  each  end  ;  and  the  whole  is  strongly  bolted  together.  Pockets  are 
left  in  the  pier  near  bottom  for  access  to  nuts,  and  these  pockets  are 
covered  by  granite  caps  or  iron  plates. 


PROJECTIONS  OF  THE  GLOBE.  475 


PEOJECTIONS    OF    THE    GLOBE. 


UNDER  the  head  of  "  Topographical  Drawing  "  are  given  plans  of  por- 
tions of  the  earth's  surface,  and  the  conventional  signs  by  which  its 
features  are  designated.  It  now  remains  to  explain  the  common  projec- 
tions by  which  meridians  and  parallels  of  latitude  are  represented  on 
charts  and  maps. 

If  the  sphere  be  projected  orthographically,  or  in  perspective,  the  rep- 
resentation will  be  correct,  but  the  parts  will  not  admit  of  measure,  except 
for  a  small  space  in  the  centre,  or  directly  beneath  the  eye,  as  will  be 
readily  understood  by  referring  to  the  principles  given  under  "  Geometrical 
Projection "  and  "  Perspective  Drawing."  Neither  of  these  projections 
are  suited,  therefore,  to  the  purposes  of  maps,  in  which  it  is  important 
that  the  relative  distance  between  different  points  in  every  position  upon 
the  map  should  be  represented  as  accurately  as  possible.  Without  going 
fully  into  the  principles  of  the  different  projections  usually  employed,  it 
will  be  considered  sufficient  for  the  present  purpose  to  explain  how  the 
meridians  and  parallels  are  projected. 

GLOBULAR  PROJECTION  OF  THE  SPHERE. — According  to  this  method,  the 
circles  of  the  sphere  should  be  represented  by  ellipses ;  but  in  practice, 
and  as  employed  in  most  school  maps,  they  are  represented  approximately 
by  circles.  fThe  following  is  the  construction  : 

To  project  a  hemisphere  (fig.  1). — Draw  two  lines,  at  right  angles  to  and 
intersecting  each  other,  from  the  point  C  of  their  intersection  as  a  centre, 
with  a  radius  equal  to  that  intended  for  the  hemisphere,  describe  a  circle, 
and  mark  the  points  N",  S,  W,  E.  N"  and  S  will  be  the  poles,  the  line  N  S 
the  central  meridian,  and  W  E  the  equator.  Divide  N  S  and  W  E  into  as 
many  equal  parts  as  there  are  degrees  or  numbers  of  degrees  to  be  rep- 
resented— in  the  figure  in  divisions  of  30° — and  meridian  and  equator 
into  6  equal  parts,  as  the  hemisphere  embraces  180°.  Commence  at 
C,  and  divide  the  half  lines  into  three  equal  parts.  Divide  the  arcs 
N  TV,  IS"  E,  S  "W,  and  S  'E,  each  into  3  equal  parts.  There  will  be 


4T6 


PROJECTIONS   OF   THE   GLOBE. 


Fig.  1. 


now  determined  3  points  in  2  parallels  of  north  and  south  latitude,  30° 
and  60°,  through  which  to  describe  the  arcs  representing  the  parallels. 

The  centre  of  these  arcs  will 
be  in  the  line  K"  S ;  describe 
the  arc,  and  with  the  same 
radius  from  a  centre  on  the 
line  N  S,  below  the  S  pole, 
describe  a  similar  arc  pass- 
ing through  the  S  30°  point 
on  the  meridian.  There- 
fore, keeping  the  steel  point 
of  the  dividers  on  the  line 
N  S,  by  trial  radii  may  be 
found  of  arcs  which  shall 
pass  through  the  points  on 
the  central  meridian  and 
on  the  circle.  "With  the 
radii  describe  arcs  for  the 
parallels  in  north  and  south 

latitude.  All  the  meridians  pass  through  the  IS"  and  S  poles,  and  through 
the  divisions  of  degrees  on  the  equator.  There  are  3  points,  therefore,  de- 
termined in  the  arc  of  each  meridian  which  may  be  described  frojn  centres 
found  by  trial  on  the  line  E  "W. 

STEBEOGEAPHIC  PROJECTION. — To  project  the  hemisphere  on  the  plane 
of  the  meridian  (fig.  2). — Draw  central  meridian,  equator,  and  circle,  as 

in  the  preceding  problem.  To  pro- 
ject the  other  meridians  (say  every 
10°),  divide  the  quadrant  N"  E  into 
9  equal  parts ;  from  S  to  these  points 
of  division,  10,  20,  30,  draw  lines  in- 
tersecting C  E  in  10,  20,  30.  These 
latter  points  are  in  the  meridians 
through  which  N  and  S  arcs  are  to 
be  described  from  centres  on  the  line 
EW. 

To  find  in  like  manner  the  3 
points  in  the  parallels  of  latitude, 
divide  the  quadrants  into  9  parts, 
80,  TO,  60,  and  through  these  points  draw  lines  to  "W  ;  the  inter- 
sections with  the  central  meridian  80,  TO,  60,  will  with  the  points  of 


PBOJECTION3   OF  THE   GLOBE. 


477 


the  quadrant  furnish  3  points  through  which  to  describe  arcs  of  paral- 
lels of  latitude. 

To  project  the  hemisphere  on  the  plane  of  the  equator  (fig.  3}. — Draw 
two  lines  at  right  angles  to  each  other ; 
describe  the  circle  and  divide  the  cir- 
cumference  as   before.     The  centre  C 
will  be  the  projection  of  1ST  or  S  pole, 
the  lines  at  right  angles  to  each  other 
will  be  meridians,  as  well  as  any  other  A[ 
diameters,  as  D  H,  F  K,  drawn  through 
some  division  of  the  circumference. 

To  project  the  parallels  of  latitude. 
— The  circle  represents  the  projection  of 
the  equator,  and  the  other  parallels 
must  be  arcs  on  the  same  centre  C,  Fis-3. 

of  which  the  radii  are  to  be  determined  by  the  intersections  of  the  line 
C  B  by  lines  drawn  from  A  to  the  divisions  of  the  circle  10,  20,  30. 

To  project  the  hemisphere  on  the  plane  of  the  horizon  for  a  given 
latitude  (fig.  4). — Draw  the  two  lines  at  right  angles  to  each  other,  and 
describe  the  circle  as  before  (fig.  4) ;  and,  to  prevent  confusion  with  the 
constructive  lines,  draw  a  similar  figure  (fig.  5).  The  circle  N  W  S  E  will 


represent  the  horizon  on  which  the  sphere  is  projected,  and  K,  "W,  S,  E  are 
the  cardinal  points.  Lay  off  W  P'  equal  to  the  given  latitude,  and  draw 
E'  P' ;  the  point  P"  in  which  it  intersects  W  C'  will  be  the  projection  of 
the  pole,  which,  laid  off  on  fig.  4,  is  represented  by  P.  Draw  the  lines 


478 


PROJECTIONS  «OF   THE   GLOBE. 


P'  C7  and  F  A'  perpendicular  to  it  at  C' ;  draw  A7  E7 :  its  intersection  A" 
with  C'  S7  will  be  a  point  in  the  arc  of  the  equator.  Transfer  this  point  to 
A  (fig.  4),  and  through  the  3  points  W,  A,  E,  describe  arc  for  the  projection 
of  the  equator.  For  other  parallels  lay  off  (say)  20"  and  40"  each  side  of 
P7 ;  draw  E'  20,  E7  40,  intersecting  C'  N7  and  C'  S7  in  a',  c',  d',  b7 ;  a7  V 
and  c7  d7  are  the  diameters  of  the  projected  parallels,  corresponding  to  50° 
and  70°  of  latitude.  Transfer  these  points  to  fig.  4,  and,  on  a  b  and  c  d  as 
diameters,  describe  circles  for  the  above  parallels. 

To  project  the  meridians. — The  lines  N  P  P  S  are  the  projections  of  the 
opposite  meridians  which  pass  through  the  N  and  S  points  of  the  horizon. 
Draw  the  tangent  S7  D,  and  extend  P7  p,  intersecting  it  at  D.  .  In  fig.  4 
take  C  B  equal  to  D  S7  (fig.  5),  and  from  B  as  centre  describe  a  circle  pass- 
ing through  P.  It  will  pass  through  AY  and  E,  and  will  be  the  projection 
of  the  meridian  of  the  place  for  which  the  projection  is  made.  Draw 
G  H  through  B  and  perpendicular  to  C  S  ;  at  B  lay  off  the  angles  which 
the  meridians  make  with  each  other  (say  15°),  B  P  15,  B  P  30  ;  the 
intersections  15,  30,  on  the  line  G  H  will  be  centres  on  which  to  de- 
scribe through  the  point  P  the  projected  meridians  required,  as  m  P  n7, 
n  P  m7,  etc. 

CONSTRUCTION  OF  MAPS  BY  DEVELOPMENT. — The  methods  of  projection 
already  explained  are  usually  confined  to  the  delineation  of  a  hemisphere  ; 
but  for  the  delineation  of  a  single  country  the  method  of  development  is 
employed,  which  exhibits  with  greater  precision  the  correct  distances  be- 
tween places,  while  for  the  purposes  of  navigation,  where  the  bearings  of 
places,  one  from  another,  must  be  correctly  and  simply  shown,  the  J/ier- 

cator's  Chart  is  used. 

It  is  obvious  that  the  surface  of 
a  sphere  cannot  be  precisely  repre- 
sented by  a  plane-surface,  but  that, 
for  small  extents,  it  may  very  nearly 
coincide  with  that  of  a  cone  or  a 
cylinder. 

Following  preceding  constructions, 
|E  let  N  W  S  E  (fig.  6)  be  the  section  of 
a  sphere  on  the  plane  of  the  meridian, 
N  S  the  axis,  and  W  E  the  diameter 
of  the  equator.  Take  any  arc,  E  F, 
and  bisect  it  at  G ;  through  G  draw 
a  tangent  intersecting  the  axis  pro- 
duced at  L.  If  the  hemisphere  re- 


w  — 


PROJECTIONS  OF  THE  GLOBE. 


4T9 


volvo  about;  the  axis  K  S,  it  will  generate  a  sphere, 
while  the  tangent  L  d  will  generate  a  conical  surface  j 
and  it  may  be  readily  seen  that  the  surface  of  the  sphere 
embraced  between  the  two  parallels  f  F  and  e  E,  and 
the  meridians  passing  through  any  two  points,  as  II 
and  G  on  the  central  parallel  g  G,  will  diner  but  little 
from  the  conical  surface  embraced  between  the  lines 
L  H  and  L  G,  which  is  developed  in  fig.  7  by  construc- 
tion explained,  pp.  102-10-i. 


Fig.  7. 


Table  showing  the  Number  of  Geographic  Miles  in  a  Degree  of  Longi- 
tude, under  each  Parallel  of  Latitude. 


Parallel 
of  Latitude. 

Geog.  Miles  in  a 
Degree. 

Parallel 
of  Latitude. 

Geog.  Miles  in  a 
Degree. 

Parallel 
of  Latitude. 

Geog.  Miles  in  a 
Degree. 

0 

o 

0 

0 

60.00 

30 

52.00 

60 

30.07 

1 

59.99 

31 

51.47 

61 

29.16 

2 

59.96 

32 

50.93 

62 

28.24 

3 

59.92 

33 

50.37 

63 

27.31 

4 

59.85 

34 

49.79 

64 

26.37 

5 

59.77 

35 

49.20 

65 

25.43 

6 

59.67 

36 

48.60 

66 

24.47 

7 

59.56 

37 

47.97 

67 

23.51 

8 

59.42 

38 

47.34 

68 

22.54 

9 

59.26 

39 

46.69 

69 

21.56 

10 

59.09 

40 

46.02 

70 

20.58 

11 

58.90 

41 

45.35 

71  - 

19.59 

12 

58.70 

42 

44.65 

72 

18.60 

13 

58.47 

43 

43.95 

73 

17.59 

14 

58.23 

44 

43.23 

74 

16.59 

15 

57.97 

45 

42.50 

75 

15.58 

16 

57.69 

46 

41.75 

76 

14.56 

17 

57.39 

47 

40.99 

77 

13.54 

18 

57.08 

48      . 

40.22 

78 

12.51 

19 

56.75 

49 

39.44 

79 

11.48 

20 

56.40 

50 

38.64 

80 

10.45 

21 

56.04 

51 

37.83 

81 

9.42 

22 

55.66 

52 

37.01 

82 

8.38 

23 

55.26 

53 

36.18 

83 

7.34 

24 

54.84 

54 

35.34 

84 

6.29 

25 

54.41 

55 

34.40 

85 

5.24 

26 

53.96 

56 

33.63 

86 

4.20 

27 
28 

53.50 
53.01 

57 

58 

32.75 

31.87 

87 
88 

3.15 
2.10 

29 
30 

52.52 
52.00 

59 
60 

30.98 
30.07 

89 
90 

1.05 

0.00 

480 


PROJECTIONS    OF   THE    GLOBE. 


Each  degree  of  latitude  is  _  always  GO  geographical  miles,  while  a 
degree  of  longitude  is  60  miles  only  at  the  equator,  and  becomes  less 
toward  the  poles. 

A  map  may  quite  accurately  present  a  portion  of  a  sphere  by  repre- 
senting it  as  the  development  of  a  cylinder  of  which  the  parallels  are 
straight  lines,  and  distant  from  each  other  on  a  scale  of  GO  miles  for  each 
degree,  and  the  central  meridian  perpendicular  to  the  parallels,  and  the 
others  inclined,  and  distant  from  the  central  one  according  to  their  posi- 
tions in  latitude.  Thus,  in  fig.  8,  which  embraces  40°  of  latitude,  and  ex- 
tends from  10°  to  40°  K  latitude  in  divisions  of  10°  each,  the  parallels 


Fig.  8. 

are  600  miles  apart,  the  meridians  on  10°  IS".  590.9  miles,  on  40°  K  460.2 
miles.  The  meridians  are  here  drawn  as  straight  lines  ;  but  it  would  be 
more  accurate  if  the  length  of  meridians  were  laid  off  on  each  parallel, 
and  curved  lines  drawn.  In  addition,  the  parallels  may  be  drawn  in  con- 
centric circles,  taking  radii  from  the  development  of  the  cone  (fig.  7). 

To  construct  a  Mercator's  Chart  (fig.  9).— Draw  two  straight  lines,  "W  E 
and  N  S,  intersecting  each  other  at  right  angles  at  C.  WE  is  the  equator, 
1ST  S  the  meridian  passing  through  the  middle  of  the  chart.  From  C  set 
off  equal  parts  on  the  equator  both  ways,  to  represent  degrees  of  longi- 


PROJECTIONS   OF    THE   GLOBE. 


481 


tude,  subdivided  into  minutes  if  the  size  of  the  chart  will  admit  of  it. 
Assuming  the  equator  as  a  scale  of  minutes  set  off  from  0  toward  N  and 
S,  the  number  of  minutes  in  the  enlarged  meridian  corresponding  to  each 


degree  of  latitude,  as  shown  by  the  table  of  meridional  parts.  Draw  lines 
parallel  to  N  S  through  the  divisions  of  the  equator  for  meridians,  and 
parallels  to  "W  E  through  the  divisions  of  1ST  S  for  parallels  of  latitude. 

To  find  the  bearing  of  any  one  place  from  another  it  is  only  necessary  to 
draw  a  straight  line  between  the  two  points,  and  observe  the  angle  it  makes 
with  the  meridians. 


Table  of  Meridional  Parts. 


Latitude. 

Meridional  Parts. 

Latitude. 

Meridional  Parts.        Latitude. 

Meridional  Parts. 

o 

0 

0 

0.00 

35 

2244.29 

70 

5965.92 

5 

300.38 

40 

2629.69 

75 

6970.34 

10 

G03.07 

45 

3029.94    - 

80 

8375.20 

15 

910.46 

50 

3474.47 

85 

10764.62 

20 

1225.14 

55 

3967.97 

90 

Infinite 

25 

1549.99 

GO 

4527.37 

30 

1888.38 

65 

5178.81 

01 


482  SPECIFICATIONS. 


SPECIFICATIONS. 

IN  the  construction  of  works,  buildings,  tools,  or  machinery,  it  is  im- 
possible to  make  every  thing  intelligible  by  plans  and  drawings  only. 
This  necessity  is  supplied .  by  written  descriptions,  called  "  specifications," 
which  form,  with  the  plans,  a  part  of  the  contract. 

The  specifications  define  the  work  to  be  done,  the  materials  of  which  it 
is  to  be  composed,  the  kind  of  workmanship  ;  exhibit  dimensions  and  parts 
not  shown  on  plan,  and  are,  in  general,  explanatory,  so  that  contractor 
or  workmen  may  execute  and  finish  agreeably  to  the  intention  of  the  de- 
signer, who  states  the  different  works  in  the  order  they  will  be  probably 
executed,  as,  if  a  building,  with  the  excavation,  then  masonry,  carpentry, 
plumbing,  painting,  etc. ;  if  machinery,  a  general  description  usually  pre- 
faces the  specification,  which  then  follows  in  some  natural  order.  In  Eng- 
lish specifications,  it  is  customary  to  make  out  tables  of  quantities  of 
materials,  on  which  the  contractor  bases  his  estimates — a  far  better  way 
than  that  here  adopted,  where  every  contractor  makes  out  his  own  bill  of 
items,  involving  an  unpaid  trouble  to  him,  and  great  liability  to  mistakes. 
A  full  bill  of  items  is  more  explanatory  of  the  work  than  any  thing  else  can 
be,  and,  in  my  own  practice,  I  have  found  that  with  it  there  is  by  far 
greater  uniformity  in  the  bids  received  from  responsible  parties. 

In  the  heading  of  the  specification,  the  kind  of  work  (building,  ma- 
chine, etc.),  where  it  is  to  be  erected  or  delivered,  and  that  it  is  to  be  done 
agreeably  to  the  specification  and  accompanying  plans  made  by  ....  (ar- 
chitect, engineer,  etc.),  should  be  stated.  Another  clause  specifies  the  party 
— usually  the  same  as  above — to  whose  approval  the  work  is  subject,  and 
to  whom  the  disputes  between  contracting  parties  are  to  be  referred.  If 
possible,  it  is  well  to  state  some  standard  of  similar  work,  to  which  "  ma- 
terials and  workmanship  shall  be  in  every  respect  equal."  A  common 
phrase  is,  to  state  that  "  all  the  several  materials  used  are  to  be  of  the  very 
best  quality,  and  all  the  work  to  be  done  in  the  best  and  most  workman- 
like manner." 

Some  provision  is  to  be  made  for  the  commencement,  finishing,  and 
delivering  of  the  work  ;  also  for  any  damages  to  persons  or  property,  that 
may  arise  during  the  execution  of  the  work,  and  for  insurance. 


APPENDIX. 


Extracts  from  the  Law  relating  to  Buildings  in  the  City  of  New  York. 

THE  footing,  or  base  course,  under  all  foundation-walls,  shall  be  of  stone  or  concrete, 
and  at  least  12"  wider  than  the  bottom  width  of  the  foundation-walls.  And  if  the 
walls  be  built  of  isolated  piers,  then  there  must  be  inverted  arches,  at  least  12"  thick, 
turned  under  and  between  the  piers,  or  2  footing  courses  of  large  stone,  at  least  10" 
thick  in  each  course.  All  foundation-walls  other  than  those  of  dwellings  shall  be  at 
least  4"  thicker  than  the  wall  next  above  them,  to  a  depth  16'  below  the  curb-level,  and 
shall  be  increased  4"  for  every  additional  5'  in  depth  below  the  said  16'.  Foundation- 
walls  in  dwelling-houses  shall  be,  below  the  basement-floor  beams,  4"  thicker  than  the 
walls  next  above  them.  By  foundation-walls  is  meant  that  portion  of  the  wall  below 
the  level  of  the  street-curb,  and  below  the  basement-floor  beams  in  dwellings,  and  depth 
shall  be  computed-from  the  curb-level  downward. 

In  all  dwelling-houses  not  above  30'  in  height,  and  not  more  than  20'  in  width,  the 
party  and  outside  walls  shall  not  be  less  than  8"  thick ;  hi  all  dwelling-houses  from  30' 
to  55'  in  height,  and  not  more  than  30'  in  width,  the  outside  and  party-walls  shall  not 
be  less  than  12"  ;  and  if  above  55',  the  walls  shall  not  be  less  than  16",  to  the  top  of  the 
2d-story  beams ;  provided  the  same  is  20'  above  the  curb,  if  not,  then  to  the  under  side 
of  the  3d-story  beams ;  and  also  provided  that  portion  of  the  12"-wall  shall  not  exceed 
40'  in  height  from  said  16"-wall. 

In  all  buildings  other  than  dwelling-houses,  not  above  30'  in  height,  and  not  more 
than  25'  in  width,  the  outside  walls  shall  not  be  less  than  8"  thick,  and  the  party-walls 
not  less  than  12" ;  if  above  30'  and  under  50'  in  height,  the  outside  walls  shall  not  be 
less  than  12",  and  the  party-walls  not  less  than  16" ;  if  above  50'  and  under  65'  in 
height,  the  outside  walls  shall  not  be  less  than  16"  to  the  height  of  the  3d-story  beams, 
and  not  less  than  12"  from  thence  to  the  top,  and  the  party-walls  not  less  than  20"  to 
the  height  of  the  2d-story  beams,  and  not  less  than  16"  from  thence  to  the  top ;  and  if 
above  65'  and  under  80'  in  height,  the  outside  walls  shall  not  be  less  than  16"  to  the 
height  of  at  least  40',  and  up  to  the  under  side  of  the  next-story  beams  above,  and  not 
less  than  12"  from  thence  to  the  top,  and  the  party-walls  not  less  than  20"  to  the  height 
of  the  3d-story  beams,  and  not  less  than  16"  from  thence  to  the  top.  In  all  buildings 
over  27'  in  width,  and  not  having  either  brick  partition-walls  or  girders,  supported  by 
columns  running  from  front  to  rear,  the  walls  shall  be  increased  an  additional  4"  in 
thickness,  to  the  same  relative  thickness  in  height  as  required  under  this  section,  for 
every  additional  10'  in  width  of  said  building ;  and  in  all  buildings  intended  or  used 


484:  APPENDIX. 

for  the  purposes  of  storing  or  keeping  heavy  merchandise  or  materials,  the  walls  shall 
be  an  additional  4"  thicker  than  above  required.  The  amount  of  materials  specified 
may  be  used  either  in  piers  or  buttresses,  provided  the  walls  between  the  same  shall  in 
no  case  be  less  than  8"  thick,  to  the  height  of  40',  and  if  over  that  height,  then  12" ; 
and  if  over  65',  16",  to  the  height  of  20'  from  the  curb. 

All  stone  walls  less  than  24"  thick  shall  have  at  least  1  header,  extending  through 
the  walls,  in  every  6  sq.  ft. ;  and  if  over  24",  shall  have  1  header  for  every  6  sq.  ft.  on 
both  sides  of  the  wall,  and  running  into  the  wall  at  least  2'.  In  every  brick  wall  every 
5th  course  of  bricks  shall  be  a  heading  course,  except  where  walls  are  faced  with  brick, 
in  which  case  every  5th  course  shall  be  bonded  into  the  backing.  In  all  walls  which 
are  faced  with  thin  ashlar,  anchored  to  the  backing,  the  backing  of  brick  shall  not  be 
less  than  8",  laid  up  in  cement  mortar,  and  shall  not  be  built  to  greater  height  than 
prescribed  for  8"-walls. 

Every  isolated  pier  less  than  6  sq.  ft.  at  the  base,  and  all  piers  supporting  a  wall 
built  of  rubble-stone  or  brick,  or  under  any  iron  beam  or  arch-girder,  or  arch  on  which 
a  wall  rests,  or  lintel  supporting  a  wall,  shall,  at  intervals  of  not  more  than  30"  in 
height,  have  built  into  it  a  bond-stone  not  less  than  4"  thick,  of  a  diameter  each  way 
equal  to  the  diameter  of  the  pier,  except  that  in  front  piers  above  the  curb,  the  bond- 
stone  may  be  4"  less  than  the  pier  in  diameter ;  the  walls  and  piers  under  all  compound 
girders,  iron  or  other  columns,  shall  have  a  bond-stone  every  30"  in  height  from  the 
bottom,  whether  said  piers  are  in  the  walla  or  not,  and  a  cap-stone  at  least  12"  in  thick- 
ness by  the  whole  size  of  the  bearing. 

All  walls  shall  be  securely  anchored  with  iron  anchors,  to  each  tier  of  beams.  The 
front,  rear,  side,  end,  and  party-walls  shall,  if  not  carried  up  together,  be  anchored  to 
each  other,  every  6'  in  their  height,  by  wrought-iron  tie-anchors,  1 J"  x  f  ",  built  into  the 
side  or  party-walls  not  less  than  16",  and  into  the  front  and  rear  walls  at  least  i  the 
thickness  of  the  front  and  rear  walls ;  and  all  stone  used  for  the  facing  of  any  build- 
ing, except  where  built  with  alternate  headers  and  stretchers,  shall  be  strongly  anchored 
with  iron  anchors,  let  into  the  stone  at  least  1".  The  side,  end,  or  party-walls,  shall  be 
anchored  at  each  tier  of  beams,  at  intervals  of  not  more  than  8'  apart,  with  f  "  x  1" 
wrought-iron  anchors,  well  built  into  the  side-walls  and  fastened  to  the  side  of  the 
beams ;  and  where  the  beams  are  supported  by  girders,  the  ends  of  the  beams  resting  on 
the  girder  shall  be  strapped  by  wrought-iron  scraps  of  the  same  size,  and  at  the  same 
distance  apart,  and  in  the  same  beam  as  the  wall-anchors. 

Compound'beams  with  cast-iron  arches  and  wrought-iron  ties,  used  to  span  open- 
ings not  more  than  10'  in  width,  shall  have  a  bearing  of  at  least  12"  by  the  thickness 
of  the  wall  to  be  supported,  and  for  every  additional  foot  of  span  over  and  above  the 
said  10',  the  bearing  shall  be  increased  1",  provided  the  same  are  supported  at  the  ends 
on  brick  or  stone,  and  on  the  front  of  any  building  where  the  supports  are  of  iron  or 
solid  cut  stone,  they  shall  be  at  least  12"  on  the  face,  and  the  width  of  the  thickness 
of  the  wall  to  be  supported,  and  shall  rest  upon  a  cut-granite  base  block  at  least  12" 
thick  by  the  full  size  of  the  bearing,  and  all  compound  beams  or  girders  used  in  any 
building  shall  be,  throughout,  of  a  thickness  not  less  than  the  thickness  of  the  wall  to 
be  supported.  All  compound  beams  shall  have  a  cast-iron  shoe  on  the  upper  side,  to 
answer  for  the  skew-back  of  a  brick  or  cut-stone  arch,  to  be  turned  over  the  same,  and 
in  no  case  less  than  8"  in  height  by  the  width  of  the  wall  to  be  supported.  Cut-stone 


APPENDIX.  485 

or  hard  brick  arches,  where  the  arch  has  not  abutments  of  sufficient  size  to  resist  its 
thrust,  may  be  turned  over  any  opening  less  than  40',  provided  they  have  skew-backs 
of  cut  stone  or  cast  or  wrought  iron,  into  which  tension-rods  are  properly  secured. 

All  chimneys  and  all  flues  in  stone  or  brick  walls,  in  any  building,  shall  be  properly 
pargetted,  or  the  joints  shall  be  struck  smooth  on  the  inside.  And  flues  or  pipes  of  a 
single  thickness  of  metal,  to  convey  heated  air  in  any  building,  shall  be  so  constructed 
as  to  have  a  thickness  of  not  less  than  1"  of  plaster  of  Paris  between  the  said  metal  and 
any  of  the  wood-work  adjoining  the  same.  No  smoke-pipe  in  any  building  shall  enter 
any  flue,  unless  the  said  pipe  shall  be  at  least  18"  from  either  combustible  floors  or  ceil- 
ings ;  and  where  smoke-pipes  pass  through  stud  or  wooden  partitions  of  any  kind,  they 
shall  be  guarded  by  either  a  double  collar  of  metal,  with  at  least  4"  air-space,  and  holes 
for  ventilation,  or  by  a  soapstone  ring,  not  less  than  3"  in  thickness  and  extending 
through  the  partition,  or  by  a  solid  coating  of  plaster  of  Paris,  3"  thick,  or  by  an 
earthenware  ring,  3"  from  the  pipe.  In  no  building  shall  any  wooden  beams  or  timbers 
be  placed  within  8"  of  any  flue.  All  wooden  timbers  in  the  party-wall  of  every  buliding 
shall  be  separated  from  the  beam  or  timber  entering  in  the  opposite  side  of  the  wall  by 
at  least  4"  of  solid  mason-work.  No  floor-beam  shall  be  supported  wholly  upon  any 
wood  partition,  but  every  beam,  except  headers  and  tail-beams,  shall  rest,  at  each  end, 
not  less  than  4"  in  the  wall,  or  upon  a  girder. 

All  exterior  cornices  and  gutters  of  all  buildings  shall  be  of  some  fire-proof  material. 
The  planking  and  sheathing  of  the  roof  of  every  building  shall  in  no  case  be  extended 
across  the  front,  rear,  side,  end,  or  party-wall  thereof. 

All  gas,  water,  steam,  or  other  pipes,  in  any  building  other  than  a  dwelling-house, 
shall  not  be  let  into  the  beams,  unless  the  same  be  placed  within  36"  of  the  end  of  the 
beams,  and  not  more  than  2"  in  depth. 

In  all  buildings,  every  floor  shall  be  of  sufficient  strength  in  all  its  parts  to  bear 
safely,  upon  every  sq.  ft.  of  its  surface,  75  Ibs. ;  and  if  used  as  a  place  of  public  assem- 
bly, 120  Ibs. ;  and  if  used  for  any  manufacturing  or  commercial  purposes,  from  150  to 
500  Ibs.  and  upward,  in  addition  to  the  weight  of  the  materials  of  which  the  floor  is 
composed ;  and  every  column  shall  be  of  sufficient  strength  to  bear  safely  the  weight 
of  the  portion  of  each  and  every  floor  depending  upon  it,  in  addition  to  the  weight 
required  as  above  to  be  supported  safely  upon  said  portions  of  said  floors.  In  all  cal- 
culations for  the  strength  of  materials  to  be  used  in  every  building,  the  proportion 
between  the  safe  weight  and  the  breaking  weight  shall  be  as  1  to  3  for  all  pieces  sub- 
jected to  a  cross  strain;  and  shall  be  as  1  to  6  for  all  vertical  supports,  and  for  tie-rods, 
and  pieces  subjected  to  a  tensile  strain. 

In  all  fire-proof  buildings,  either  brick  walls  with  wrought-iron  beams,  or  cast  or 
wrought-iron  columns  with  wrought-iron  beams,  must  be  used  in  the  interior.  The 
metarcolumns  shall  be  planed  true  and  smooth  at  both  ends,  and  shall  rest  on  cast-iron 
bed-plates  and  have  cast-iron  caps,  also  planed  true.  If  brick  arches  are  used  between 
the  beams,  the  arches  shall  have  a  rise  of  at  least  1J"  to  each  foot  of  span  between  the 
beams.  All  arches  shall  be  at  least  4"  thick.  Arches  over  4'  span  shall  be  increased  in 
thickness  toward  the  haunches  by  additions  of  4"  in  thickness  of  brick  ;  the  first  addi- 
tional thickness  shall  commence  at  2}'  from  the  centre  of  the  span  ;  the  second  addition 
6V  from  the  centre  of  the  span;  and  the  thickness  shall  be  increased  thence  4"  for  every 
additional  4'  of  span  toward  the  haunches. 


486 


APPENDIX. 


tween  the  bearings  of  continuous  shafts, 
subject  to  no  transverse  strain  except  from 
their  own  weights. 


Distance  between  bearings, 

Diameter  of 
shaft 

in  feet. 

in  inches. 

If  of 
wrought  iron. 

If  of  steel. 

1 

12.27 

12.61 

2 

15.46 

15.89 

3 

17.70 

18.19 

4 

19.48 

20.02 

5 

20.99 

21.57 

6 

22.30 

22.92 

7 

23.48 

24.13 

8 

24.55 

25.23 

9 

25.53 

26.24 

10 

26.44 

27.18 

11 

27.30 

28.05 

12 

28.10 

28.88 

Extracts  from  "  Formulas  and  Tables  for  the  Shafting  of  Mills  and 
Factories"  by  Mr.  J.  B.  Francis,  Journal  of  the  Franklin  Institute. 

"Shafts  for  transmitting  power  are  subject  to  two  forces,  viz. :  transverse  strain  and 
torsion.     In  shafts  of  wrought  iron  or  steel,  in  which  the  bearings  are  not  very  near  to 

each  other,  the  weight  of  the  shaft  itself  will  pro- 
Table  of  the  greatest  admissible  distances  be-     v    • 

duce  an  inadmissible  amount  of  deflection  when- 
ever this  distance  exceeds  a  certain  amount,  which 
varies  with  the  material  and  diameter  of  the  shaft. 
"In  practice,  long  shafts  are  scarcely  ever  en- 
tirely free  from  transverse  strains ;  however,  in  the 
parts  of  long  lines  which  have  no  pulleys  or  gears, 
with  the  couplings  near  the  bearings,  the  interval 
between  the  bearings  may  approach  the  distances 
given  in  the  table.  Near  the  extremities  of  a 
line,  the  distances  between  the  bearings  should  be 
less  than  are  given  in  the  table.  The  last  space 
should  not  exceed  60$  of  the  distance  there  given, 
the  deflection  in  that  space  being  much  greater 
than  in  other  parts  of  the  line. 

"In  factories  and  workshops,  power  is  usually 
taken  off  from  the  lines  of  shafting,  at  many  points, 
by  pulleys  and  belts.  "When  the  machines  to  be 
driven  are  below  the  shaft,  there  is  a  transverse  strain  on  the  shaft,  due  to  the  weight 
of  the  pulley  and  tension  of  the  belt,  in  addition  to  the  transverse  strain  due  to  the 
weight  of  the  shaft  itself.  When  the  power  is  taken  off  horizontally  on  one  side,  the 
tension  of  the  belt  produces  a  horizontal  transverse  strain;  and  the  weight  of  the 
pulley  acts  with  the  weight  of  the  shaft,  to  produce  a  vertical  transverse  strain.  "When 
the  machinery  to  be  driven  is  placed  above  the  floor,  to  which  the  shaft  is  hung  in  the 
story  below,  the  transverse  strain  produced  by  the  tension  of  the  belt  is  in  the  opposite 
direction  to  that  produced  by  the  weight  of  the  pulley  and  shaft.  To  transmit  the  same 
power,  the  necessary  tension  of  a  belt  diminishes  in  proportion  to  its  velocity;  conse- 
quently, with  pulleys  of  the  same  diameter,  the  transverse  strain  will  diminish  in  the  same 
ratio  as  the  velocity  of  the  shaft  increases.  In  cotton  and  woollen  factories  with  wooden 
floors,  the  bearings  are  usually  hung  on  the  beams,  which  are  usually  about  8'  apart;  and 
a  minimum  size  of  shafting  is  adopted  for  the  different  classes  of  machinery  which  has 
been  determined  by  experience  as  the  least  that  will  withstand  the  transverse  strain. 
This  minimum  is  adopted  independently  of  the  size  required  to  withstand  the  torsional 
strain  due  to  the  power  transmitted ;  if  this  requires  a  larger  diameter  than  the  mini- 
mum, the  larger  diameter  is.  of  course,  adopted.  In  some  of  the  large  cotton-factories  in 
this  neighborhood  (Lowell,  Mass.),  in  which  the  bearings  are  about  8'  apart,  a  minimum 
diameter  of  1^"  was  formerly  adopted  for  the  lines  of  shafting  driving  looms.  In  some 
mills  this  is  still  retained,  in  others  2|"  and  2Ty  have  been  substituted.  In  the  same 
mills,  the  minimum  size  of  shafts  driving  spinning  machinery  is  from  2J"  to  2y|".  In 
very  long  lines  of  small  shafting,  fly-wheels  are  put  on  at  intervals,  to  diminish  the  vibra- 
tory action  due  to  the  irregularities  in  the  torsional  strain. 


APPENDIX. 


4:87 


"The  following  ttible  gives  the  power  which  can  be  safely  carried  by  shafts  making 
100  revolutions  per  minute.  The  power  which  can  be  carried  by  the  same  shafts  at  any 
other  velocity  may  be  found  by  the  following  simple  rule : 

"  Multiply  the  power  given  in  the  table  ~by  the  number  of  revolutions  made  by  the 
shaft  per  minute ;  divide  the  product  l>y  one  hundred;  the  quotient  will  ~be  the  power 
which  can  ~be  safely  carried" 


DIA3IETER 
IN 
INCHES. 

;orse-power  which  can  be  safelv  carried 
by  shafts  for  prime  movers  and  gears, 
well  supported  by  bearings,  and  making 
100  revolutions  per  minute  ;  if  of 

Horse-power  which  can  be  safely  transmit- 
ted by  shafts  making  100  revolutions  per 
minute,  in  which  the  transverse  strain,  if 
any,  need  not  be  considered  ;  if  of 

Wrought  iron. 

Steel. 

Cast  iron. 

Wrought  iron. 

Steel. 

Cast  iron. 

1.00 

1.00 

1.60 

0.60 

2.00 

3.20 

1.20 

1.25 

1.95 

3.12 

1.17 

3.90 

6.24 

2.34 

1.50 

3.37 

5.39 

2.03 

6.74 

10.78 

4.06 

1.75 

5.36 

8.58 

3.22 

10.72 

17.16 

6.44 

2.00 

8.00 

12.80 

4.80 

16.00 

25.60 

9.60 

2.25 

11.39 

18.22 

6.83 

22,78 

36.44 

13.66 

2.50 

15.62 

24.99 

9.37 

31.24 

49.98 

18.74 

2.75 

20.80 

33.28 

12.48 

41.60 

66.56 

24.96 

3.00 

27.00 

43.20 

16.20 

54.00 

86.40 

32.40 

3.25 

34.33 

54.93 

20.60 

68.66 

109.86 

41.20 

3.50 

42.87 

68.59 

25.72 

85.74 

137.18 

51.44 

3.75 

52.73 

84.37 

31.64 

105.46 

168.74 

63.28 

4.00 

64.00 

102.40 

38.40 

128.00 

204.80 

76.80 

4.25 

76.77 

122.83 

46.06 

153.54 

245.66 

92.12 

4.50 

91.12 

145.79 

54.67 

182.24 

291.58 

109.34 

4.75 

107.17 

171.47 

64.30 

214.34 

342.94 

128.60 

5.00 

125.00 

200.00 

75.00 

250.00 

400.00 

150.00 

5.25 

144.70 

231.52 

86.82 

289.40 

463.04 

173.64 

5.50 

166.37 

268.19 

99.82 

332.74 

532.38 

199.64 

5.75 

190.11 

304.18 

114.06 

380.22 

608.36 

228.12 

6.00 

216.00 

345.60 

129.60 

432.00 

691.20 

259.20 

6.25 

244.14 

390.62 

146.49 

488.28 

781.24 

292.98 

6.50 

274.62 

439.39 

164.78 

549.24 

878.78 

329.56 

6.75 

307.55 

492.08 

184.53 

615.10 

984.16 

369.06 

7.00 

343.00 

548.80 

205.80 

686.00 

1097.60 

411.60 

7.25 

381.08 

609.73 

228.65  • 

762.16 

1219.46 

457.30 

7.50 

421.87 

674.99 

253.13 

843.74 

1349.98 

506.26 

7.75 

465.48 

744.77 

279.29 

930.96 

1489.54 

558.58 

8.00 

512.00 

819.20 

307.20 

1024.00 

1638.40 

614.40 

8.25 
8.50 

561.52 
614.12 

898.43 
982.59 

336.91 
368.47 

1123.04 

1228.24 

1796.86 
1965.18 

673.82 
736.94 

8.75 

669.92 

1071.87 

401.95 

1339.84 

2143.74 

803.90 

9.00 

729.00 

1166.40 

437.40 

1458.00 

2332.80 

874.80 

9.25 
9.50 
9.75 

791.45 
857.37 
926.86 

1266.32 
1371.79 
1482.98 

474.87 
514.43 
556.12 

1582.90 
1714.74 
1853.72 

2532.64 
2743.58 
2965.96 

949.74 
1028.86 
1112.24 

10.00 

1000.00 

1600.00 

600.00 

2000.00 

3200.00 

1200.00 

488  APPENDIX. 

From  Mr.  Francises  notes  we  extract  the  following  on  the  power  re- 
quired to  drive  various  machinery : 

"  At  Lowell  we  commonly  reckon  for  No.  14  cotton  goods  33  dead-throstle  spindles 
and  all  the  other  machinery  to  a  horse-power.  The  Rhode  Island  people,  I  believe, 
reckon  GO  spindles  to  a  horse-power  on  print-cloths,  ring-warp  and  mule-filling.  In 
1850,  Mr.  John  Newell  weighed  the  power  used  to  drive  the  machinery  in  a  mill  of 
12,544  dead-throstle  spindles  making  No.  14  yarn,  including  all  the  machinery,  but  not 
the  shafting,  which  he  found  to  be,  ....  175,860  Ibs.  ft.  per  second. 
Shafting  (estimated), 43,965 


Total, 319,825  « 

or  31.4  horse-power  per  spindle. 

"  The  proportion  for  each  room  he  made  as  follows : 

Picking-room, 0.12583 

Card           "             0.18557 

Spinning    " 0.47801 

Drawing    "             0.03821 

Weaving    " 0.17338 


Total, 1.00000 

"  From  experiments  by  Mr.  Newell  on  Rock  Bottom  Flannel  Mill,  Stow,  Mass. : 

1  picker, 1154.18  Ibs.  ft.  per  second. 

7  sets  of  cards, 7769.79       "  " 

15  jacks  (2859.  sp.), 3628.07 

18  plain  looms, , 

28  twilled  looms, 

1  dresser,  2  warpers,  2  spoolers,  1  small  pump, 

1  rotating  fulling-mill, 

2  washers, 

1  napper, • 


13$  for  geering, 


19346.44  =  35.17  HP.' 


The  Uses  of  Profile  and  Cross-section  Paper.— At  page  365  mention 
is  made  of  the  construction  of  profile  and  cross-section  paper,  and  their 
use  with  reference  to  railway  plots  ;  but  they  are  now  of  much  more  ex- 
tended application.  There  are  many  facts  involving  two  factors  or  consid- 
erations, which  can  be  expressed  graphically  in  the  form  of  a  profile,  the 
abscissas  representing  one,  and  the  ordinates  the  other ;  and  consequently 


APPENDIX. 


489 


more  readily  by  the  paper  already  prepared  for  the  purpose.  Thus,  the 
fluctuations  of  gold,  of  coal,  and  the  iron  trade,  are  represented  by  ab- 
scissas of  time,  and  ordinates  of  price,  or  of  amount  of  sales  ;  the  sanitarian 
represent  time  as  above,  and  the  number  of  deaths  by  ordinates  ;  the  rail- 
way engineer,  speed  of  train  by  one,  and  resistance  by  the  other,  etc. ; 
while  the  architect  and  mechanic  find  a  very  valuable  use  of  cross-section 
paper  in  designing  or  copying,  making  the  squares  scales  of  parts. 

Fig.  1  represents  the  path  of  float  in  a  wooden  flume  or  channel,  taken 
from  the  last  edition  of  Francis  Lowell's  "  Hydraulic  Experiments."  The 
cut  was  copied  directly  on  the  wood,  and  is  therefore  reversed.  The 


1  (\ 

^ 

J-~v 

-o 

Oi> 

X 

->~v 

^ 

.X 

~ 

V. 

- 

' 

/ 

J 

0 

, 

0, 

3 

, 

'5 

! 

! 

^ 

Q 

Q 

JE 

•-•" 

s 

^;. 

, 

1 

V 

> 

c 

Fig.  1. 


width  of  the  cut  represents  the  width  of  the  flume,  each  abscissa  being  1 
ft. ;  the  ordinates  are  the  speeds  of  float  in  divisions  of  0.1  ft.  per  second ; 
the  o  o  are  the  floats  in  their  observed  path,  and  speed ;  and  the  curved 
line  the  average  velocity  in  the  different  threads  of  the  stream. 

Fig.  2  is  from  Clarke's  "  Railway  Machinery."     The  abscissas  repre- 


Miles  per  hour. 


'      § 


20  30  40  50  GO-    fl' 


0, 

1 

• 

-. 

"i- 

i 

•-* 

•(} 

1  • 

-> 

-, 

1 

^^^^ 

i 

a 

> 

•v. 

3 

^ 

00 

ii 

•v 

li 

90 

Fig.  2. 


sent  the  speed  in  miles  per  hour ;  the  ordinates  the  Ibs.  per  ton  resistance 
of  a  100-ton  train. 


490 


APPENDIX. 


Fig.  3  is  made  up  from  time-table  of  ]ST.  Y.  &  "N.  H.  K.,  showing  the 
movement  of  trains,  two  from  ]STew  York  and  two  from  Kew  Haven,  the 
abscissas  being  cut  off  on  a  scale  of  miles  for  each  station,  the  ordinates 
being  a  scale  of  hours  : 


s:s  s 


Jn    HI  uiif  i  1  il  li 

^3    ,3  «    ^    §     8       |    g|     |       ft     o    2      =;= 


Fig.  3. 


IISTDEX. 


A    CANTHUS,  273. 

-£A_  Acoustics,  principles  of,  292. 

Alhambra,  276. 

Angle,  greatest,  under  which  objects  can  be  seen, 
386. 

Angles  taken  by  compass,  how  laid  off  by  pro- 
tractor, 358. 

Antae,  240. 

Aqueduct.  Brooklyn,  438,  431 ;  Croton,  439 ;  iron 
pips  across  Harlem  River,  439,  440. 

Aqueducts,  2  GO. 

Arcades,  259. 

Arch,  dimension  of,  bridges,  457 ;  depth  of  key- 
stone, 458 ;  radius  at  crown,  segmental,  ellip- 
tical, 488 ;  effect  of  unequal  loads  on,  458, 
459 ;  height  of  spandrel  backing,  459  ;  hori- 
zontal thrust,  459. 

Arches,  terms  applied  to  parts  of,  215  ;  rules  to 
determine  the  depth  at  crown  of,  216 ;  trian- 
gular-headed, round-headed,  stilted,  horseshoe, 
pointed,  complex,  foiled,  263,  264 ;  four-centred 
or  Tudor,  266. 

Architrave*,  28%,  242. 

Ashlar  (see  MASONRY). 

Asphalt  pavement,  453. 

Atlantes,  259. 

Avenue,  450. 

Axles,  size  of,  129. 

BATTER  (see  MASONRY). 
Balcony,  284. 
Ballast  for  railway,  455. 
Balusters,  279,  280. 
Base  and  surbase  of  rooms,  243. 
Basilicas,  294. 
Bay-windows,  281. 
Beams  and  girders,  124-128  ;  \vooden,  216-218; 

collar,  tie  (see  ROOF). 
Bearings  for  shafts,  142,  145,  146. 
Belting,  strain  on,  155  ;  strength  of,  156. 


Bevels  (see  GEERS). 

Bevel-wheels,  projections  of,  179,  181 ;  skew  bev- 
els, 181 ;  isometrical  view  of,  410. 

Blinds,  242. 

Boarding,  vertical,  281. 

Boilers,  evaporation  of,  136  ;  construction  of, 
467-471. 

Bolts,  size  and  proportions  of,  190  ;  thread,  nuts, 
191 ;  washers,  192. 

Bond  (see  MASONRY). 

Boulevards,  450. 

Bracing,  principles  of,  230,  231. 

Bridges,  girder  or  frame,  460  ;  piers  of,  455,  456  ; 
arch,  457-460 ;  skew,  460 ;  suspension,  465,  466. 
I  Brushes  for  tinting,  333. 

Buttresses,  266. 

Buildings,  expression  of  purpose  in,  301 ;  prin- 
ciples of  design  of,  of  construction  of,  essential 
conditions  of  good,  truth  in  expression,  present 
taste,  material  for,  309-311. 

Building  act,  extracts  from  New  York,  483-485. 

Byzantine  architecture,  260. 

CABLING,  257. 
Camera  lucida,  35. 

Campanile,  268. 

Cam-punch  and  shear,  frame  of,  192. 

Canals,  sections  of,  431  ;  for  transportation,  430; 
for  the  supply  of  mills,  431,  432. 

Carriage-way,  452. 

Caryatides,  259. 

Catch-basins,  307  ;  drawing  of,  trap  of,  448. 

Cathedral,  pi.  LXXVI. 

Centre  of  gravity,  113  ;  of  surfaces,  of  solids,  1 14. 

Central  plane,  387. 

Cesspools,  245,  307. 

Chimneys,  245,  246,  472 ;  chimney-tops,  282. 

Chancel,  294. 

Churches,  292,  294;  Church  of  the  Holy  Sep- 
ulchre, 260. 


-±92 


INDEX. 


Ciiiquecento,  278. 

Coal,  consumption  of,  468. 

Coffer-dam,  419 ;  for  Susquehanna  bridge  piers, 
419,  420. 

Colors,  preparation  of,  for  tints,  381  ;  applica- 
tion of,  381,  382. 

Coloring,  finished,  339  ;  color  suited  to  different 
materials,  examples  of,  and  method  of  laying 
on,  selection  of  colors,  340-348. 

Columns,  cast-iron,  122 ;  spaces  between,  258  ; 
super  columnation,  258. 

Compass,  use  of,  in  surveying,  357 ;  plot  of  a 
survey  by,  358. 

Compassers  or  dividers,  11,  12;  triangular,  27; 
proportional,  28  ;  beam,  29  ;  portable,  30 ; 
tubular,  large  screw,  31. 

Composite  order,  examples  of,  257. 

Concrete,  use  of,  421. 

Conduits  for  water-supply,  438  ;  cross-section  of 
main  for,  Brooklyn  Water-works,  438,  439  ; 
Croton,  439. 

Cone,  development  of  surface  of,  103 ;  of  rays 
of  light,  385. 

Cones,  90 ;  penetrations  of,  93-99  ;  pulleys,  154. 

Conic  sections,  construction  of,  90,  91. 

Connecting  rods,  drawings  of,  and  proportions 
of,  197, 198. 

Contours,  drawing  of  hills  by,  353 ;  map  of  por- 
tion of  city  of  London,  383  ;  town  and  country 
maps  drawn  with,  384. 

Conventional  signs,  topographical,  349,  350. 

Correction  of  errors  in  surveys,  or  balancing  ex- 
amples of,  359,  360. 

Copying  glass,  368  ;  of  large  plans,  369. 

Cornice,  234  ;  interior,  243,  244. 

Cornices,  Gothic,  Norman,  English,  262. 

Cornish  Pumping  Engine,  202,  462. 

Corinthian  order,  examples  of,  256. 

Corbels,  262. 

Couplings,  face,  148  ;  sleeve,  screw,  149 ;  clamp, 
box,  horned,  150;  clutch,  151;  friction  cone, 
152. 

Country-house,  drawings  of,  pi.  LXV.-LXIX. ; 
designing  of,  235. 

Crank,  drawing  of,  table  of  relative  size  of  eyes 
in,  196. 

Crockets,  277. 

Culvert,  isometrical  view  of,  410. 

Curbs,  452. 

Curves',  definition  of,  3-5 ;  construction  of,  and 
problem  on  circles,  ellipse,  parabola,  hyperbo- 
la, cycloid,  epicycloid,  involute,  spiral,  49,  79 ; 
catenary,  117;  helix,  99;  eccentric,  185. 

Cylinder,  projections  of,  84 ;  penetrations  of,  92, 
97  ;  development  of  surface  of,  103. 


DADO,  243. 
Dam  across  Connecticut  River,  at  Holyoke, 
425;  across  Herrimack  River,  at  Lowell,  425, 
426 ;  across  Mohawk,  at  Cohocs,  426,  427 ; 
across  Croton  River,  427  ;  for  pondage,  428; 
across  the  Furens,  in  France,  429. 

Day's  work,  131. 

Design  of  buildings,  principles  of,  30. 

Development  of  surfaces,  103. 

Dimensions  fcr  different  spans  of  truss,  463. 

Dome,  260. 

Domes  and  vaults,  264-266. 

Doors,  terms  applied  to  parts  of,  proportions  of, 
and  drawings  of,  239,  240;  of  stores,  287. 

Doorways,  Norman,  261  ;  circular-headed,  Tudor- 
arched,  Gothic,  gabled,  perpendicular,  Byzan- 
tine, Saracenic,  renaissance,  Florentine,  Vene- 
tian, and  Roman,  271,  272. 

Doric  order,  examples  of,  254,  255. 
|  Dowels,  218. 

j  Drainage,  306  ;  of  road-beds,  454,  455  ;  and  sewer- 
!      rage  of  part  of  city  of  London,  383. 

Drawing,  first  design,  working,  246  ;  paper  for 
tinting,  333  ;  table,  board,  paper,  35. 

Drawings,  skeleton,'  partial,  outline,  201,  202  ; 
working,  202. 

Drums,  154. 

EAVES,  finish  of,  227. 
Eccentrics,  projections  of,  185-189. 
Edifices,  purposes  of,  shown  by  appropriate  sign 

in  topography,  351. 
Egg  and  dart,  273. 
Elizabethan  style,  278. 
Embankment,  section  of  Thames,  421. 
Engineer,  object  of,  416. 
Engineering,  definition  of  civil,  415. 
Entablature,  253,  257,  259. 
Entasis,  254. 
Equilibrium  of  arches,  266 ;  of  the  polygon  of 

rods,  117;  of  framework,  118. 
1  Evaporation  of  different  fuels,  135. 
Extrados  (see  ARCHES) 

Exterior  of  buildings,  material,  style,  color,  299- 
|      300. 

tracery,  265. 

Feed-water,  temperature  of,  468. 
Field-book,  355. 
Finishing  of  plans,  369. 
Fireplace,  244,  246. 

Floor?,  218;  fire-proof,  219;  of  stores,  287. 
Florentine  school  of  architecture,  272. 
Flues,  244 ;    for  heating   and   ventilating,  304 ; 
dampers  to,  305. 


INDEX. 


493 


Flumes,  construction  of,  4-38. 

Footwalk,  452. 

Forces,  parallel,  110;  inclined,  111;  resultants, 
components,  parallelogram  of,  112  ;  application 
of,  113. 

Formula?,  Neville's  and  Dupius's,  for  cast-iron 
pipes,  442  ;  for  the  discharge  of  pipes,  442. 

Foundations,  209,  210;  for  dams,  427;  of  con- 
duit, Brooklyn  Water-works,  439. 

Frame  of  wooden  house,  terms  applied  to  parts 
of,  220. 

Frames,  application  of  iron  to,  192-194. 

Framing,  119  ;  wood,  as  applied  to  buildings,  216 ; 
flooring,  bridging,  headers,  trimmers,  and  tail- 
beams,  217  ;  girders  and  joists,  217. 

Friction,  coefficient  of,  115;  limiting  angle  of  re- 
sistance to,  116  ;  Morin's  experiments  on,  JIG; 
of  surfaces,  117. 

Fuel,  table  of  evaporative  powers  of  different 
kinds  of,  135. 

Furnaces,  hot-air,  303,  304. 

GALLOWS,  frame,  of  drawbridge,  461. 
Gas,   supply,    449  ;    service-pipe,   mains, 
450. 

Gauging  flo\v  of  streams,  428,  429. 

Geers,  spur,  mitres,  157  ;  internal,  rack  geer  and 
pinion,  bevel,  trundle-face,  leader,  follower,  size 
of  bevel  gecrs,  158;  skewed  bevels,  160;  pitch 
and  form  of  teeth  of,  160-166 ;  to  calculate 
strength  of  teeth  of,  167;  epicycloid  teeth, 
169 ;  preparation  and  use  of  templates,  172, 
173 ;  involute  teeth,  175  ;  projections  of  wheels, 
175-184. 

Geological  features,  representations  of,  382. 

Geological  map  of  California,  pi.  XCVII. 

Geometrical  definitions,  1-6 ;  problems  on  lines, 
42-49  ;  arcs  and  circles,  49-55  ;  parallel,  55  ; 
circles,  and  rectilinear  figures,  55,  67  ;  on  the 
ellipse,  parabola,  hyperbola,  cycloid,  and  epi- 
cycloid, 68-76  ;  the  involute,  78  ;  spiral,  79  ; 
projections,  80 ;  constructions,  85,  100. 

Gib  and  cotter,  197. 

Girder,  126,  232  ;  form  adopted  for  cast  iron, 
by  Mr.  Cubett,  462  ;  of  Crystal  Palace,  229. 

Globular  projections  of  the  sphere,  475. 

Golden  Gate,  elevation  and  section  of  engines 
of,  204,  205. 

Gothic  church,  294. 

Grade  of  roads,  454. 

Greek  temples,  plans  of,  258. 

Grillage,  219. 

Groins,  265  (see  ARCHES). 

Ground  plane,  387. 

Guilloche,  273. 


TTAND-RAIL  (see  STAIRS). 

-Ll  Hangers,  147,  148. 

Headgates  of  canals,  at  Cohoes,  430  ;  of  flumes, 
438. 

Heating,  301 ;  of  stores,  287  ;  of  schools,  289. 

Helix,  projections  of,  99-101. 
j  Hermes  pillars,  259. 

High  Bridge,  Harlem  River,  260. 

Hills  expressed  by  brush-drawing,  354 ;  repre- 
sented under  an  oblique  light,  380 ;  how 
shaded  in  topographical  drawings,  379;  topo- 
graphical, 351-352. 

Hoistways,  286. 

Hoisting  apparatus  at  Cohoes  headgates,  430. 

House,  elevations  of,  from  Holly's  country-houses, 
pi.  LXV. ;  country-residence,  Downing,  LXVI., 
LXVIII.,  LXIX. ;  cottage,  Gervase  Wheeler, 
LXVII. ;  villa,  Upjohn,  LXIX. ;  elevation  of, 
English  basement,  280,  LXII. 

Houses,  plan  and  elevations  of,  pi.  XLVII.-LI. ; 
various  plans,  249,  250 ;  city,  basement,  250. 

Housing?,  193. 

Hooks,  forms  of,  192. 

Hot  water,  heating  by,  304. 

Hydrostatic  press,  105,  192. 

TNCLIXED  plane,  105,  109. 

-J-  Ink,  China,  318 ;  China,  to  draw  lines  in,  40  ; 
inkles,  41. 

Internal  wheel  and  pinion  projections,  183,  184. 

Intrados  (see  ARCHES). 

Ionic  order,  examples  of,  255,  256. 

Iron  fronts,  288  ;  Works,  Althause,  312. 

Isometrical  view  of  Victoria  and  Regent  Street 
sewers,  Thames  embankment,  423;  drawing, 
principles  and  examples  of,  405-414  ;  projec- 
tions of  a  cube,  405-408 ;  of  a  prism,  408 ;  of 
curved  lines,  409  ;  of  a  bevel-wheel,  409  ;  of  a 
a  pillow-block,  410 ;  of  a  culvert,  410 ;  of  a 
boiler,  truss-bridge,  roof-truss,  school-house, 
411. 

"TACK-SCREW,  drawing  of,  192. 
*J    Joints  in  plates,  463. 
Journals,  size  of,  133-141. 

T  ECTURE-ROOMS,  292. 

-L^  Legislative  halls,  292 ;  Chamber  of  French 
Deputies,  298 ;  Mr.  Mills  and  Reid  on  form 
of,  298 ;  House  of  Representatives,  Washing- 
ton, 298,  299. 

Lettering,  370 ;  various  kinds  of  letters  and  type, 
370-374 ;  irregular,  of  roads,  of  curved  lines, 
spacing  of  letters,  374 ;  examples  of,  375  ;  of 
tinted  drawing,  382. 


494 


Lever,  105-107, 108-113. 

Light,  how  diffused,  reflected,  31D ;  direction  of, 
on  plans  and  elevations,  314 ;  in  topographical 
drawings,  352 ;  how  it  is  supposed  to  fall  on 
topographical  drawings,  369  ;  in  churches, 
296. 

Lightning,  308. 

Line  drawing?,  138. 

Lines  of  series,  secants,  tangents,  semitangents, 
rhumbs,  longitude,  21 ;  of  chords,  23;  of  poly- 
gons, 24 ;  of  sines,  lines,  tangents,  26 ;  scale 
of,  of  circles  of  planes,  28;  dotted  full  and 
broken,  42;  boundary,  how  designated,  370; 
of  horizon,  387. 

Locks  of  canals,  433 ;  plans  and  details  of,  434- 
437. 

Locomotive,  201,  202  ;  sections  of  boiler  of,  204  ; 
boiler,  468. 

Louis  Quinze  style,  279. 

"jy/TACHIKES,  location  of,  198;  examples  of 

•JJ-L  two  weaving-rooms  in  plans  and  sections, 
199,  200. 

Man-holes  of  sewers,  446 ;  covers  of,  447. 

Mansard,  or  gambrel  roof,  227. 

Mantels,  244. 

Maps  by  development,  478. 

Marbling,  how  done,  382. 

Masonry,  terms  applied  to,  212-215. 

Materials,  mechanical  properties  of,  120;  resist- 
ance to  compressure,  121-123  ;  to  tension, 
124;  to  transverse  strains,  124-128;  to  de- 
tention, to  torsion,  128,  129 ;  for  exterior  of 
edifices,  299 ;  style  of,  299. 

Mechanical  powers,  105-107. 

Mechanical  work,  130;  unit  of,  130;  terms  of, 
131 ;  of  men  and  animals,  131. 

Mercator's  chart,  481. 

Meridian?,  370. 

Metals,  strength  of,  121  ;  topographical  marks 
for,  351. 

Moisture,  table  of  grains  in  one  cubic  foot  of  air, 
302. 

Motion,  transmission  of,  158. 

Motors,  131. 

Mortar,  215. 

Mouldings,  names  of,  and  construction  of,  251, 
252;  Gothic,  261;  arch,  262;  hood,  262; 
scroll,  263. 


-*~™    Xew  Haven,  map  of  harbor  and  city  of, 

pi.  XCIV. 
Xew  World,  frame  of  engine,  193  ;  working-beam, 

194. 


/~\RDERS  of  architecture,  253. 

V-'  Ornaments,  constructive,  decorative,  Gre- 
cian, Roman,  Symbolic,  Byzantine,  Saracenic, 
Gothic,  Renaissance,  Elizabethan,  272-279. 

TDACIFIC,  frame  of  engine,  193. 

-L  Palace,  of  Diocletian,  260;  Crystal,  origin 
and  description  of,  299. 

Pantheum,  column,  254. 

Paper,  drawing,  35 ;  tracing,  36 ;  stretching,  37  ; 
mounting,  38 ;  cleaning,  40 ;  preparing,  for  tints, 
381 ;  profile,  and  cross  section,  365,  489  ;  tra- 
cing, 368. 

Parapets,  277. 

Partitions,  219 ;  bridging,  219  ;  drawing  of,  246. 

Pavements,  452,  453. 

Pentograph,  33. 

Perspective,  angular,  388  ;  parallel,  389-401 ; 
drawing,  385  ;  linear  and  aerial,  386. 

Pier,  crib,  quarantine  for  port  of  New  York,  423, 
424;  pile,  455;  trestle,  456;  form  of  ends 
Susquehanna  river  bridge,  456. 

Pile  piers,  455. 

Pile-driver,  417  ;  Nasmyth,  417. 

Piling  sheet,  how  driven ;  material  for,  417,  418. 

Piles,  how  used,  416;  material  of,  417;  hollow, 
cast-iron,  how  driven,  example  from  Harlem 
bridge,  418,  419. 

Pillars,  sections  of  Gothic,  pi.  LVI.,  261,  262, 

Pillow-block,  146  ;  isometrical  view  of,  410. 

Pinnacles,  267-269. 

Pipes,  cast  iron,  plans  and  section  of,  Brooklyn 
water,  442 ;  formula  for  strength  of,  442 ;  ex- 
amples of  branches  and  stems,  444 ;  specifica- 
tion for  Brooklyn,  444;  table  of  weights  of, 
445 ;  gas,  weight  of,  449. 

Piston  and  rods,  473,  474. 

Pitch  (see  GEERS,  SCREWS,  ROOFS). 

Plane  of  the  picture,  387. 

Planing-machine,  frame  of,  192. 

Plot,  how  to  close,  359  ;  of  railway  line,  364. 

Plotting,  355  ;  offsets,  361 ;  offsets  scale,  362 ;  of 
roads,  canal  railway  survey,  362 ;  from  a  sur- 
vey by  compass,  by  theodolite,  359. 

Point  of  view,  station,  387 ;  of  distance,  vauish- 

!      ing  points,  388. 

Power,  absolute,  effective,  107 ;  water,  steam,  132 ; 
horse,  133  ;  required  to  drive  machinery,  488. 

{  Prism,  projections  of,  88-90. 

Privies,  245. 

i  Profile,  of  road,  454 ;  of  railway  line,  365  ;  paper 

|      scales  for,  365  ;  paper,  365,  489. 

|  ProtrUctor,  19  ;  circular,  31. 

j  Pulleys,  153  ;  line,  154  ;  width  of  face,  155;  fast 

!     and  loose,  156;  position  of,  157. 


495 


Purlincs  (see  ROOF). 
Pyramids,  projections  of,  85-88. 

RACK  and  pinions,  projections  of,  182. 
Racks,  438. 
Rafters  (see  ROOFS). 
Railway,  curves,  designation  of  and  plotting  of, 

3G3,  364 ;  width  of,  ballast  for,  length  of  cross- 
ties,  455. 
Rain-fall,  429 ;  rain-shed,  how  much  can  be  cal 

culated,  430. 
Ranges,  244. 
Reflection  of  objects   in  water,  in  perspective, 

402. 
Reservoirs,  size  of,  440 ;  Ridgewood,  New  Cro- 

ton,  440  ;  division  bank  in  Croton,  441. 
Resistance  of  wheel-carriage?,  of  sled,  455. 
Road,  McAdam,  Telford,  Central  Park,  450-454. 
Romanesque  architecture,  260. 
Romanesque  church,  294. 
Roman  school  of  architecture,  272. 
Roofs,  drawing  of  various,  221-230 ;  terms  applied 

to  parts  of,  pitch  of,  bridging  of  rafters  of,  221 ; 

form  of  foot  of  rafter  of,  222 ;  application  of 

cast  iron  to  sheer  and  abutting-plates  of,  size 

and  proportion  of  different  numbers  of,  222; 

determination  of  strain  on  different  parts  of, 

223,  224  ;  iron,  elevation  of,  228. 
Roof-truss,  isometrical  view  of,  40. 
Rooms,  size  and  proportions  of,  234;  dining,  235  ; 

parlors,  235  ;   pantries,  passages,  236 ;  details 

of  parts  of,  237  ;  finish  of,  243. 
Rubble  (see  MASONRY). 
Ruler,  common,  7;  parallel,  10. 

SAFES,  287. 
Scale,  selection  of,  17  ;  paper,  17;  diagonal, 
18;  plotting,  verneir,  plain,  19;  plain,  double- 
sectoral,  22;  to  form,  23;  Marquois's,  26;  in 
plotting,  355  ;  house-lots,  farm  surveys,  Eng- 
lish Tithe  Commission  maps,  355  ;  State 
surveys,  harbor  charts,  decimal  system  of  rail- 
road surveys,  canal  maps,  English  standing  or- 
ders, 356  ;  U.  S.  engineer  service,  357 ;  to  be 
adopted,  328  ;  of  shades  for  slopes  of  ground, 
353  ;  to  be  drawn  on  maps,  370 ;  in  perspec- 
tive, 392. 

School-houses,  288 ;  plans  and  elevation  of,  pi. 
LXXIV. ;  on  the  requirements  of,  location  of 
desks,  maps,  black-boards,  259;  heating  of, 
plan  of  New  York  City  schools,  289-291 ;  re- 
quirements of  Sunday,  295  ;  isometrical  vie 
of,  411. 

Sector,  22-26. 

Screw,  110;  projections  of,  189,  190. 


Sewers,  306 ;  principle  in  establishing,  445 ;  incli- 
nation to  be  given  to,  445 ;  rule  to  determine 
size  of,  446  ;  value  of,  pipe,  brick,  to  determine 
area  of  egg-shaped,  446 ;  Union  Avenue,  Brook- 
lyn, 446  ;  angle  of  branches,  447 ;  Victoria  and 
Regent  Street,  423. 

Shade-lines,  83 ;  French  system,  85  ;  topographi- 
cal, 349;  on  a  cylinder,  318;  on  a  reversed 
cone,  319  ;  of  a  ring,  325  ;  surfaces  in,  329. 

Shading,  methods  of,  328  ;  of  a  cylinder  by  flat 
tints,  330;  of  a  prism,  of  a  cylinder,  by  soft- 
ened  tints,  332 ;  examples  of  finished,  335,  336. 

Shades  and  shadows,  313. 

Shadows  cast  by  a  straight  line  on  a  vertical  wall, 
314 ;  upon  a  curved  surface,  315 ;  of  a  circle  on 
a  vertical  plane,  317 ;  on  surfaces  inclined  to 
each  other,  317;  cast  by  a  pyramid,  318  ;  by 
and  on  a  cylinder,  319;  by  one  prism  on  an- 
other, 320 ;  by  a  cylinder  on  a  prism,  321 ;  on 
the  interior  of  a  hollow  cylinder,  321 ;  steam 
cylinder,  321-223 ;  in  interior  of  sphere,  324 ; 
by  the  sphere  on  a  plane,  325 ;  upon  the  sur- 
face of  grooved  pulleys,  326,  327 ;  upon  the 
surfaces  of  screws  and  nuts,  327;  in  topo- 
graphical drawings,  378,  379  ;  projections  of, 
in  perspective,  402,  403. 

Shafts,  torsion  of,  129. 

Shafting,  wooden,  cast-iron,  fastened,  tubular,  1st, 
2d,  and  3d  movers,  137 ;  strain  on,  138 ;  size 
of,  138,  485;  load  on,  139;  torsional  strain  on, 
140 ;  size  of  wrought-iron,  141 ;  upright,  144. 

Sheds  for  wood  or  coal,  24C. 

Sheet-piling,  210. 

Shipper,  152. 

Shutters,  revolving,  287. 

Skeleton  drawing,  Mandoley  and  Field's  marine 
engine,  201. 

Sketch,  rough  topographical,  355. 

Skew  backs  (see  ARCHES). 

Skylight  in  apse,  296. 

Slopes  of  ground  by  scale  of  shades,  German  sys- 
tem, 353. 

Spandrels  (see  ARCHES). 

Specifications,  extracts  from,  for  Thames  embank- 
ment, 421-423  ;  extracts  from,  for  quarantine 
pier,  424,  425  ;  extracts  from,  for  lock,  New 
York  State  canals,  435-437 ;  extract  from,  for 
construction  of  new  Croton  reservoir,  441, 442 ; 
form  of,  482. 

Sphere,  development  of  surface  of,  103, 104 ;  pro- 
jections and  penetrations  of,  94-99 ;  projec- 
tions of,  475-481. 

Spires,  267-269. 

Spurs  (see  GEERS);  projections  of,  175-179. 

Stables,  284. 


496 


INDEX. 


Stairs,  terms  applied  to  parts  of,  proportions  of, 

and  laying  out  of,  237-239,  246. 
Steam,  effective  pressure   on   piston,  expansion, 

133  ;  pressure  at  different  densities,  134 ;  table 

of  pressures,  temperature,  and  volumes,  135  ; 

heating  by,  304;  cylinder,  321 ;  engines,  466. 
Steps,  142,  145. 
Stereographic  projection,  476. 
Stirrup-irons,  217. 
Stop-cocks,  elevations,  sections,  and  plans  of  48''- 

gate,  Brooklyn  Water-works,  203. 
Store,  elevation  of,  by  J.  B.  Snook,  pi.  LXXII. ; 

cast-iron  front  of,  LXXIII. 
Stores,  height  of,  236  ;  and  warehouses,  286,  pi. 

LXXI. 
Stoves,  302. 

Streets,  450 ;  pave  of,  452. 
String-courses,  262. 
Stuffing-box,  473. 
Surveys,  how  conducted,  357. 
Susquehanna,  framing  of  engine,  194. 
Suspension  bridges,  465. 
Sweeps,  1 ;  variable  curves,  10. 
Symbols  as  ornament,  275. 

rpEETH  of  wheel  (see  GEERS). 

J-    Temple,  of  Concord,  260 ;  at  Talavera,  260. 

Tenons  and  mortices,  dra\vings  of,  220. 

Templates  (see  GEERS). 

Theatres,  292;  requirements  of,  297  ;  dimensions 
of  several,  298. 

Thread  (see  SCREWS  AND  BOLTS). 

Timber,  dimensions  of,  for  frames  of  dwelling- 
houses,  220. 

Tinting,  methods  of,  ?28-335. 

Tinted  topographical  drawings,  conventional  tints 
and  signs,  378. 

Titles,  flourishes  around,  382 ;  of  plans,  376, 
377. 

Toggle-joint,  105. 

Topographical  drawing,  349  ;  in  single  tint  by 
brush,  380. 

Towers,  267-269. 

Tracing  of  drawings,  368. 

Tractive  force  of  a  horse,  455. 

Traps,  siphon,  catch-basin,  S,  307. 

Transept,  294. 

Transfer  of  plots,  368. 


Truss  (see  ROOF)  ;  by  suspension-rod,  232. 

Trussing  girders  or  beams,  218. 

Trestles,  456. 

T  square,  9. 

Tubular  boilers,  468. 

Tudor  flower,  277. 

Turbine  wheel,  plan  and  section  of  turbine  and 
wheel  post,  Tremont  Manufacturing  Company, 
Lowell,  Mass.,  205  ;  description  of,  rules  for 
the  proportioning  of  parts  of,  drawings  of 
guides  in  disk,  and  floats  in  wheel,  206. 

Tuscan  order,  examples  of,  254. 

~T~TARIATION  of  magnetic  needle,  357. 

V      Vaults  and  domes,  264-266. 
Vaults,  to  protect,  from  moisture,  286. 
Venetian  school  of  architecture,  272. 
Ventilation,  301-306 ;  of  sewers,  447. 
Ventilators,  302,  305. 
Verge-boards,  281. 
Viaducts,  260. 
Volutes,  256. 
Voussoirs  (see  ARCHES). 

WALLS,  terms  applied  to  parts  of,  212,  213  ; 
construction  of,  bond,  batter,  brick,  stone, 
returning,  offsets,  212,  213  ;  walls  of  buildings, 
London  Building  Act,  213  ;  of  canal,  432,  433 ; 
enclosing,  266. 

Water-closets,  245,  287. 

Water  supply  for  dwellings,  pipes,  size  and  kinds 
of,  308. 

Water-wheel,  wooden,  138. 

Weaving-room,  198-200. 

Wedge,  109. 

Wheel  and  axle,  186. 

Wheels,  toothed  (see  GEERS);  water,  138;  tur- 
bine, 205. 

Windows,  terms  applied  to  parts  of,  proportions 
of,  and  drawings  of,  240-242,  246;  Roman- 
esque, Norman,  lancet,  trained,  decorated, 
quatrefoil,  flamboyant,  perpendicular,  square- 
headed,  269-271. 

Window-jambs,  261. 

Woods,  strength  of,  121. 

Working-beam,  dimensions  and  drawing  of,  195, 
196. 

Worm  wheel  and  screw,  projections  of,  183. 


THE    E2O). 


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